["1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian Ta c\u00f3 A# \u00bbI = (0; 3; 0), K# H\u00bb = \u00c5 \u2212 a yH \u2212 0; zH \u2212 b\u00e3 xH ; . 2 2 \uf8f1a \uf8f4xH = Do #\u00bb c\u00f9ng ph\u01b0\u01a1ng #\u00bb n\u00ean \uf8f4 = 2 = 0) AI KH \uf8f4 c(c \uf8f2 yH \uf8f4 = b\u00b7 \uf8f4 2 \uf8f4\uf8f3zH ab Suy ra H ; c; . 22 T\u1eeb OH = 13 suy ra a2 + c2 + b2 = 169 2 4 44 (1) M\u1eb7t kh\u00e1c, HI = OH = 13 \u21d2 a \u22129 2 b \u22121 2 169 (2) = (3) 22 + (c \u2212 3)2 + 24 T\u1eeb (1) v\u00e0 (2) suy ra a2 + c2 + b2 = a \u22129 2 b \u22121 2 4 42 2 + (c \u2212 3)2 + . Do \u0111\u00f3, 9a + b + 6c = 91 M\u1eb7t kh\u00e1c, AM = (a \u2212 9; 0; \u22121), AN = (\u22129; 0; b \u2212 1). A, M, N th\u1eb3ng h\u00e0ng \u21d2 a\u22129 = \u22121 \u21d4 (a \u2212 2)(b \u2212 1) = 9 \u22129 b\u22121 \u21d4 ab \u2212 a \u2212 9b + 9 = 9 \u21d4 ab \u2212 a \u2212 9b = 0 \u21d4 a(b \u2212 1) = ab \u21d4 a = b 9b \u22121 9b 81b T\u1eeb (3) suy ra 9 \u00b7 b \u22121 + b + 6c = 91b \u2212 1 + b + 6c \u21d4 b2 + 80b + 6c = 91 b\u22121 \u21d4 6c = 91 \u2212 b2 + 80b = \u2212b2 + 11b \u2212 91 b\u22121 b\u22121 \u21d4 c = \u2212b2 + 11b \u2212 91 6(b \u2212 1) Ta c\u00f3 a2 + 4c2 + b2 = 169 \u21d4 9b 2 + 4 \u00c5 \u2212b2 + 11b \u2212 91 \u00e32 + b2 = 169 b\u22121 6(b \u2212 1) \u21d49.81b2 + b4 + 121b2 + 8281 \u2212 22b3 + 182b2 \u2212 2002b + 9b2(b \u2212 1)2 = 169.9.(b \u2212 1)2 \u21d4729b2 + b4 + 121b2 + 8281 \u2212 22b3 + 182b2 \u2212 2002b + 9b4 \u2212 18b3 + 9b2 = 1521b2 \u2212 3042b + 1521 \u21d410b4 \u2212 40b3 \u2212 480b2 + 1040b + 6760 = 0 \u221a \uf8ee \u221a 9 1+3 3 \u221a b=1+3 3\u21d2a= \u221a =9+ 3 \uf8ef 3 3\u221a \u21d4\uf8ef \u221a \u221a \uf8ef 1\u22123 3 9 1\u22123 3 = 9 \u2212 3. \uf8f0 \u21d2 a= \u221a \u22123 3 b= Tr\u01b0\u1eddng h\u1ee3p: \u221a \u221a Khi \u0111\u00f3, A# M\u00bb = \u221a \u21d2 AM = 2. A# N\u00bb = \u2212 9; a =\u221a9 + 3; b = 1 \u221a+ 3 3. 3; 0; \u22121 0; 3 3 \u221a\u21d2 AN = \u221a108. Do \u0111\u00f3, AM.AN = 2. 108 = 12 3. \u221a\u221a Tr\u01b0\u1eddng hA#\u1ee3Mp\u00bb2=: a = 9\u221a\u2212 3; b = 1 \u2212 3 3. Khi \u0111\u00f3, A# N\u00bb = \u2212 3; 0; \u2212\u221a1 \u21d2 AM = 2\u221a. \u2212 9; 0; \u22123 3 \u21d2 AN = 108. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 498 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u221a\u221a Do \u0111\u00f3, AM.AN = 2. 108 = 12 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 83 (C\u00e2u 47 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(\u22122; 1; 2) v\u00e0 \u0111i qua \u0111i\u1ec3m A(1; \u22122; \u22121). X\u00e9t c\u00e1c \u0111i\u1ec3m B, C, D thu\u1ed9c (S) sao cho AB, AC, AD \u0111\u00f4i m\u1ed9t vu\u00f4ng g\u00f3c v\u1edbi nhau. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i t\u1ee9 di\u1ec7n ABCD c\u00f3 gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t b\u1eb1ng A 72. B 216. C 108. D 36. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t AB = a, AC = b, AD = c th\u00ec ABCD l\u00e0 t\u1ee9 di\u1ec7n vu\u00f4ng \u0111\u1ec9nh D P Mc C A, n\u1ed9i ti\u1ebfp m\u1eb7t c\u1ea7u (S). aA Khi \u0111\u00f3 ABCD l\u00e0 t\u1ee9 di\u1ec7n \u0111\u1eb7t \u1edf g\u00f3c A c\u1ee7a h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt t\u01b0\u01a1ng \u1ee9ng c\u00f3 c\u00e1c c\u1ea1nh AB, AC, AD v\u00e0 \u0111\u01b0\u1eddng ch\u00e9o AA l\u00e0 \u0111\u01b0\u1eddng k\u00ednh N Ib c\u1ee7a c\u1ea7u. Ta c\u00f3 a2 + b2 + c2 = 4R2. 1 1 a2b2c2. X\u00e9t V = VABCD = abc \u21d4V2 = 36 6 M\u00e0 a2 + b2 + c2 \u221a \u21d4 \u00c5 a2 + b2 + c2 \u00e33 a2b2c2 \u21d4 \u00c5 4R2 \u00e33 B E\u221a 3 3 a2b2c2 3 36 \u00b7 V 2 \u21d4 V R3 \u00b7 4 3 \u221a3 27 V\u1edbi R = IA = 3 3. V\u1eady Vmax = 36. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 84 (C\u00e2u 41 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I (\u22121; 0; 2) v\u00e0 \u0111i qua \u0111i\u1ec3m A (0; 1; 1). X\u00e9t c\u00e1c \u0111i\u1ec3m B, C, D thu\u1ed9c (S) sao cho AB, AC, AD \u0111\u00f4i m\u1ed9t vu\u00f4ng g\u00f3c v\u1edbi nhau. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i t\u1ee9 di\u1ec7n ABCD c\u00f3 gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t b\u1eb1ng A 8 B 4. C 4 D 8. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t AD = a, AB = b, AC = c, G\u1ecdi M , N l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung D N \u0111i\u1ec3m BC, AD. Qua M k\u1ebb \u0111\u01b0\u1eddng th\u1eb3ng d song song v\u1edbi AD, qua A N d\u1ef1ng \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi AD c\u1eaft d t\u1ea1i I. Khi \u0111\u00f3 I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n ABCD. Ta c\u00f3 \u221a R = IA = 3. \u221a b2 + c2 a I M AM = ; IM = B 22 \u21d2 R2 = IA2 = a2 + b2 + c2 = 3. 4 \u221a C \u00c1p d\u1ee5ng b\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy ta c\u00f3 a2 + b2 + c2 \u2265 3 3 a2b2c2 \u21d2 abc \u2264 8. Suy ra VABCD = 1 \u2264 1 \u00b7 8 = 4 abc 6 . 6 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 499 S\u0110T: 0905.193.688","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian \u0104 C\u00e2u 85 (C\u00e2u 46 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2\u221a01\u00e492). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : \u22122 = x2 + y2 + \u00c4 3. C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau \u0111i\u1ec3m z A(a; b; c) (a, b, c l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean) thu\u1ed9c m\u1eb7t ph\u1eb3ng (Oxy) sao cho c\u00f3 \u00edt nh\u1ea5t hai ti\u1ebfp tuy\u1ebfn c\u1ee7a (S) \u0111i qua A v\u00e0 hai ti\u1ebfp tuy\u1ebfn \u0111\u00f3 vu\u00f4ng g\u00f3c v\u1edbi nhau ? A 12. B 4. C 8. D 16. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I \u00c4 \u221a\u00e4 v\u00e0 b\u00e1n k\u00ednh R = \u221a A \u2208 (Oxy) \u21d2 A(a; b; 0). 0; 0; 2 3; \u0110\u1ec3 c\u00f3 \u00edt nh\u1ea5t hai ti\u1ebfp tuy\u1ebfn qua A th\u1ecfa m\u00e3n b\u00e0i to\u00e1n th\u00ec ta c\u00f3 hai tr\u01b0\u1eddng h\u1ee3p \u221a TH1: A \u2208 (S) \u21d4 IA = R = 3. TH2: A \u2208\/ (S), khi \u0111\u00f3 \u0111\u1ec3 t\u1ed3n t\u1ea1i hai ti\u1ebfp tuy\u1ebfn vu\u00f4ng g\u00f3c nhau th\u00ec A h\u00ecnh n\u00f3n sinh ra b\u1edfi c\u00e1c ti\u1ebfp tuy\u1ebfn v\u1ebd t\u1eeb A ph\u1ea3i c\u00f3 g\u00f3c \u1edf \u0111\u1ec9nh kh\u00f4ng nh\u1ecf h\u01a1n 90\u25e6. T\u1ee9c l\u00e0 M\u00f7AN \u2265 90\u25e6\u221a\u21d4 M\u2019AI \u2265 45\u221a\u25e6 NM s\u221ain3M\u2019\u2265A\u221aI 2\u2265\u21d422IA\u21d4\u2264II\u221aMA6.\u2265 2 I \u21d4 2 \u21d4 IA 2 Do \u0111\u00f3, y\u00eau c\u1ea7u b\u00e0i to\u00e1n x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi \u221a\u221a 3 \u2264 IA \u2264 6 \u21d4 3 \u2264 IA2 \u2264 6 \u21d4 3 \u2264 a2 + b2 + 2 \u2264 6 \u21d4 1 \u2264 a2 + b2 \u2264 4. Do a, b \u2208 Z n\u00ean ta x\u00e9t c\u00e1c tr\u01b0\u1eddng h\u1ee3p sau N\u1ebfu a = 0 th\u00ec b \u2208 {\u00b11, \u00b12} N\u1ebfu b = 0 th\u00ec a \u2208 {\u00b11, \u00b12} \u00aea = \u00b11 N\u1ebfu a = 0 v\u00e0 b = 0 th\u00ec b = \u00b11. V\u1eady c\u00f3 12 \u0111i\u1ec3m A th\u1ecfa m\u00e3n \u0111\u1ec1 b\u00e0i. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 86 (C\u00e2u 47 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u: (S) : x2 + y2 + (z + 1)2 = 5. C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau \u0111i\u1ec3m A(a; b; c)(a, b, c l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean) thu\u1ed9c m\u1eb7t ph\u1eb3ng (Oxy) sao cho c\u00f3 \u00edt nh\u1ea5t hai ti\u1ebfp tuy\u1ebfn c\u1ee7a (S) \u0111i qua A v\u00e0 hai ti\u1ebfp tuy\u1ebfn \u0111\u00f3 vu\u00f4ng g\u00f3c nhau? A 20. B 8. C 12. D 16. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 500 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN I I E AH F \u221a M\u1eb7t c\u1ea7u (S) : x2 + y2 + (z + 1)2 = 5 c\u00f3 t\u00e2m I(0; 0; \u22121) v\u00e0 c\u00f3 b\u00e1n k\u00ednh R = 5 A(a; b; 0) \u2208 (Oxy), G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AI \u21d2 I \u00c5a; b ; \u22121\u00e3 22 2 G\u1ecdi E, F l\u1ea7n l\u01b0\u1ee3t l\u00e0 hai ti\u1ebfp \u0111i\u1ec3m c\u1ee7a ti\u1ebfp tuy\u1ebfn \u0111i qua A sao cho AE \u22a5 AF . \u00c5a b 1\u00e3 Ta c\u00f3: E, F c\u00f9ng thu\u1ed9c m\u1eb7t c\u1ea7u (S ) \u0111\u01b0\u1eddng k\u00ednh IA c\u00f3 t\u00e2m I ; ; \u2212 , b\u00e1n k\u00ednh 22 2 1\u221a R = a2 + b2 + 1. 2 \u0110\u1ec1 t\u1ed3n t\u1ea1i E, F th\u00ec hai m\u1eb7t c\u1ea7u (S) v\u00e0 (S ) ph\u1ea3i c\u1eaft nhau suy ra |R \u2212 R | \u2264 II \u2264 |R + R | \u221a 1\u221a 1\u221a \u221a 1\u221a \u21d4 5 \u2212 a2 + b2 + 1 \u2264 a2 + b2 + 1 \u2264 5 + a2 + b2 + 1 \u221a \u221a2 2 2 \u21d4 5 \u2264 a2 + b2 + 1 \u21d4 a2 + b2 \u2265 4(1) G\u221a\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a I tr\u00ean (AEF ) khi \u0111\u00f3 t\u1ee9 gi\u00e1c AEHF l\u00e0 h\u00ecnh vu\u00f4ng c\u00f3 c\u1ea1nh AE = HF = AI2 \u2212 5. Ta c\u00f3 IH2 = R2 \u2212 HF 2 = 5 \u2212 (AI2 \u2212 5) = 10 \u2212 AI2 \u2265 0 \u21d4 a2 + b2 + 1 \u2264 10 \u21d4 a2 + b2 \u2264 9(2) T\u1eeb (1) v\u00e0 (2) ta c\u00f3 4 \u2264 a2 + b2 \u2264 9 m\u00e0 a, b, c \u2208 Z n\u00ean c\u00f3 20 \u0111i\u1ec3m th\u1ecfa b\u00e0i to\u00e1n. C\u00e1ch kh\u00e1c: M IA N \u221a M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(0, 0, \u22121) b\u00e1n k\u00ednh R = 5. Ta c\u00f3 d(I(Oxy)) = 1 < R \u21d2 m\u1eb7t c\u1ea7u (S) c\u1eaft m\u1eb7t ph\u1eb3ng(Oxy). \u0110\u1ec3 c\u00f3 ti\u1ebfp tuy\u1ebfn c\u1ee7a (S) \u0111i qua A \u21d4 AI \u2265 R(1). C\u00f3 A(a, b, c) \u2208 (Oxy) \u21d2 A(a, b, 0), IA = a2 + b2 + 1. Qu\u1ef9 t\u00edch c\u00e1c ti\u1ebfp tuy\u1ebfn \u0111i qua A c\u1ee7a (S) l\u00e0 m\u1ed9t m\u1eb7t n\u00f3n n\u1ebfu AI > R v\u00e0 l\u00e0 m\u1ed9t m\u1eb7t ph\u1eb3ng n\u1ebfu AI = R. Trong tr\u01b0\u1eddng h\u1ee3p qu\u1ef9 t\u00edch c\u00e1c ti\u1ebfp tuy\u1ebfn \u0111i qua A c\u1ee7a (S) l\u00e0 m\u1ed9t m\u1eb7t n\u00f3n g\u1ecdi AM, AN l\u00e0 hai ti\u1ebfp tuy\u1ebfn sao cho A, M, I, N \u0111\u1ed3ng ph\u1eb3ng. T\u1ed3n t\u1ea1i \u00edt nh\u1ea5t hai ti\u1ebfp tuy\u221a\u1ebfn c\u1ee7a (S) \u0111i qua A v\u00e0 hai ti\u1ebfp tuy\u1ebfn \u0111\u00f3 vu\u00f4ng g\u00f3c v\u1edbi nhau khi v\u00e0 ch\u1ec9 khi M\u00f7AN \u2265 90\u25e6 \u21d4 IA \u2264 R 2(2). T\u1eeb (1), (2) \u21d2 4 \u2264 a2 + b2 \u2264 9. V\u00ec a, b \u2208 Z \u00aea2 = 0 \u00aea2 = 9 \u00aea2 = 4 \u00aea2 = 0 \u00aea2 = 1 \u00aea2 = 4 \u00aea2 = 4 \u21d2 b2 = 9 ho\u1eb7c b2 = 0 ho\u1eb7c b2 = 0 ho\u1eb7c b2 = 4 ho\u1eb7c b2 = 4 ho\u1eb7c b2 = 1 ho\u1eb7c b2 = 4. B\u1ed1n h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u1ea7u ti\u00ean c\u00f3 hai nghi\u1ec7m, ba h\u1ec7 sau c\u00f3 4 nghi\u1ec7m suy ra s\u1ed1 \u0111i\u1ec3m A th\u1ecfa m\u00e3n l\u00e0 4 \u00b7 2 + 3 \u00b7 4 = 20. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 501 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng B\u00c0I 2. PH\u01af\u01a0NG TR\u00ccNH M\u1eb6T PH\u1eb2NG \u0104 C\u00e2u 1 (C\u00e2u 1 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). C\u00f3 bao nhi\u00eau c\u00e1ch ch\u1ecdn hai h\u1ecdc sinh t\u1eeb m\u1ed9t nh\u00f3m g\u1ed3m 34 h\u1ecdc sinh? A 234. B A234. C 342. D C324. \u0253 L\u1eddi gi\u1ea3i. M\u1ed7i c\u00e1ch ch\u1ecdn hai h\u1ecdc sinh t\u1eeb m\u1ed9t nh\u00f3m g\u1ed3m 34 h\u1ecdc sinh l\u00e0 m\u1ed9t t\u1ed5 h\u1ee3p ch\u1eadp 2 c\u1ee7a 34 ph\u1ea7n t\u1eed n\u00ean s\u1ed1 c\u00e1ch ch\u1ecdn l\u00e0 C234. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 2 (C\u00e2u 2 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng (P ) : x + 2y + 3z \u2212 5 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 A #n\u00bb1 = (3; 2; 1). B #n\u00bb3 = (\u22121; 2; 3). C #n\u00bb4 = (1; 2; \u22123). D #n\u00bb2 = (1; 2; 3). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) : x + 2y + 3z \u2212 5 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb2 = (1; 2; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 3 (C\u00e2u 15 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng (P ) : 3x + 2y + z \u2212 4 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 A #n\u00bb3 = (\u22121; 2; 3). B #n\u00bb4 = (1; 2; \u22123). C #n\u00bb2 = (3; 2; 1). D #n\u00bb1 = (1; 2; 3). \u0253 L\u1eddi gi\u1ea3i. M\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) l\u00e0 #n\u00bb = (3; 2; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 4 (C\u00e2u 12 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng (P ) : 2x + 3y + z \u2212 1 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 A #n\u00bb1 = (2; 3; \u22121). B #n\u00bb3 = (1; 3; 2). C #n\u00bb4 = (2; 3; 1). D #n\u00bb2 = (\u22121; 3; 2). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) : 2x + 3y + z \u2212 1 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (2; 3; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 5 (C\u00e2u 2 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng (P ) : 2x + y + 3z \u2212 1 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 A #n\u00bb4 = (1; 3; 2). B #n\u00bb1 = (3; 1; 2). C #n\u00bb3 = (2; 1; 3). D #n\u00bb2 = (\u22121; 3; 2). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) : 2x + y + 3z \u2212 1 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 n#\u00bb3 = (2; 1; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 6 (C\u00e2u 1 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : x + 2y + 3z \u2212 1 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P )? A #n\u00bb3 = (1; 2; \u22121). B #n\u00bb4 = (1; 2; 3). C #n\u00bb1 = (1; 3; \u22121). D #n\u00bb2 = (2; 3; \u22121). \u0253 L\u1eddi gi\u1ea3i. T\u1eeb ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) suy ra m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng l\u00e0 #n\u00bb4 = (1; 2; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 502 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 7 (C\u00e2u 2 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 y + 3z + 1 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P ) ? C n#\u00bb2 = (2; \u22121; 3). D n#\u00bb3 = (2; 3; 1). A n#\u00bb1 = (2; \u22121; \u22123). B n#\u00bb4 = (2; 1; 3). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 y + 3z + 1 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 n#\u00bb2 = (2; \u22121; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 8 (C\u00e2u 1 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 3y + z \u2212 2 = 0. V\u00e9c-t\u01a1 n\u00e0o sau \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9ct\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P ). D #n\u00bb4 = (2; 1; \u22122). A #n\u00bb3 = (\u22123; 1; \u22122). B #n\u00bb2 = (2; \u22123; \u22122). C #n\u00bb1 = (2; \u22123; 1). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 v\u00e9c-t\u01a1 #n\u00bb1 = (2; \u22123; 1) l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P ). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 9 (C\u00e2u 2 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 4x + 3y + z \u2212 1 = 0. V\u00e9c-t\u01a1 n\u00e0o sau \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P )? A #n\u00bb4 = (3; 1; \u22121). B #n\u00bb3 = (4; 3; 1). C #n\u00bb2 = (4; \u22121; 1). D #n\u00bb1 = (4; 3; \u22121). \u0253 L\u1eddi gi\u1ea3i. (P ) : 4x + 3y + z \u2212 1 = 0. V\u00e9c-t\u01a1 #n\u00bb3 = (4; 3; 1) l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P ). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 10 (C\u00e2u 4 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : x\u22123 y+1 z+2 = \u22122 = . Vec-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t 4 3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d? C #u\u00bb2 = (4; \u22122; 3). D #u\u00bb1 = (3; 1; 2). A #u\u00bb3 = (3; \u22121; \u22122). B #u\u00bb4 = (4; 2; 3). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb2 = (4; \u22122; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 11 (C\u00e2u 19 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (\u03b1) : 2x + 4y \u2212 z + 3 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (\u03b1)? A #n\u00bb1 = (2; 4; \u22121). B #n\u00bb2 = (2; \u22124; 1). C #n\u00bb4 = (\u22122; 4; 1). D #n\u00bb3 = (2; 4; 1). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (\u03b1) : 2x + 4y \u2212 z + 3 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb1 = (2; 4; \u22121). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 12 (C\u00e2u 15 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (\u03b1) : 2x \u2212 3y + 4z \u2212 1 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (\u03b1)? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 503 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng A n#\u00bb3 = (2; \u22123; 4). B n#\u00bb2 = (2; 3; \u22124). C n#\u00bb1 = (2; 3; 4). D n#\u00bb4 = (\u22122; 3; 4). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (\u03b1) \u0111\u00e3 cho c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 n#\u00bb3 = (2; \u22123; 4). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 13 (C\u00e2u 12 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng (\u03b1) : 2x \u2212 y + 3z + 5 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (\u03b1)? B #n\u00bb4 = (2; 1; \u22123). C #n\u00bb2 = (2; \u22121; 3). D #n\u00bb1 = (2; 1; 3). A #n\u00bb3 = (\u22122; 1; 3). \u0253 L\u1eddi gi\u1ea3i. M\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (\u03b1) l\u00e0 #n\u00bb2 = (2; \u22121; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 14 (C\u00e2u 1 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (\u03b1) : x \u2212 2y + 4z \u2212 1 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (\u03b1)? A #n\u00bb3 = (1; \u22122; 4). B #n\u00bb1 = (1; 2; \u22124). C #n\u00bb2 = (1; 2; 4). D #n\u00bb4 = (\u22121; 2; 4). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (\u03b1) : x \u2212 2y + 4z \u2212 1 = 0 c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb3 = (1; \u22122; 4). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 15 (C\u00e2u 23 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 3x \u2212 y + 2z \u2212 1 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P )? D #n\u00bb(P ) = (3; 1; \u22122). A #n\u00bb(P ) = (\u22123; 1; 2). B #n\u00bb(P ) = (3; \u22121; 2). C #n\u00bb(P ) = (3; 1; 2). \u0253 L\u1eddi gi\u1ea3i. M\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) l\u00e0 #n\u00bb = (3; \u22121; 2). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 16 (C\u00e2u 13 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : \u2212 2x + 5y + z \u2212 3 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P )? A #n\u00bb2 = (\u22122; 5; 1). B #n\u00bb1 = (2; 5; 1). C #n\u00bb4 = (2; 5; \u22121). D #n\u00bb2 = (2; \u22125; 1). \u0253 L\u1eddi gi\u1ea3i. M\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) l\u00e0 #n\u00bb2 = (\u22122; 5; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 17 (C\u00e2u 7 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y + 2z \u2212 3 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P )? A #n\u00bb3 = (1; 2; 2). B #n\u00bb1 = (1; \u22122; 2). C #n\u00bb4 = (1; \u22122; \u22123). D #n\u00bb2 = (1; 2; \u22122). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y + 2z \u2212 3 = 0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bbP = (1; \u22122; 2). Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 504 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 18 (C\u00e2u 15 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 2x + 4y \u2212 z \u2212 1 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P )? A #n\u00bb2 = (2; \u22124; 1). B #n\u00bb1 = (2; 4; 1). C #n\u00bb3 = (2; 4; \u22121). D #n\u00bb4 = (\u22122; 4; 1). \u0253 L\u1eddi gi\u1ea3i. M\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P ) l\u00e0 #n\u00bb = (2; 4; \u22121). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 19 (C\u00e2u 13 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 3y + 4z \u2212 1 = 0 c\u00f3 m\u1ed9t vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 A n#\u00bb4 = (\u22121; 2; \u22123). B n#\u00bb3 = (\u22123; 4; \u22121). C n#\u00bb2 = (2; \u22123; 4). D n#\u00bb1 = (2; 3; 4). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 3y + 4z \u2212 1 = 0 c\u00f3 m\u1ed9t vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 n#\u00bb2 = (2; \u22123; 4). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 20 (C\u00e2u 47 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng (Oxz) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A z = 0. B x + y + z = 0. C y = 0. D x = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (Oxz) \u0111i qua \u0111i\u1ec3m O(0; 0; 0) v\u00e0 nh\u1eadn #j\u00bb = (0; 1; 0) l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn n\u00ean ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t ph\u1eb3ng (Oxz) l\u00e0 y = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 21 (C\u00e2u 20 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A(3; 0; 0), B(0; 1; 0) v\u00e0 C(0; 0; \u22122). M\u1eb7t ph\u1eb3ng (ABC) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 xyz xy z xyz x yz A 3 + \u22121 + 2 = 1. B 3 + 1 + \u22122 = 1. C ++ = 1. D \u22123 + 1 + 2 = 1. 312 \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng ph\u1eb3ng qua 3 \u0111i\u1ec3m A(a; 0; 0), B(0; b; 0), C(0; 0; c), abc = 0, c\u00f3 d\u1ea1ng l\u00e0 xyz + + = 1. abc N\u00ean ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng qua 3 \u0111i\u1ec3m A(3; 0; 0), B(0; 1; 0) v\u00e0 C(0; 0; \u22122) l\u00e0 xy z 3 + 1 + \u22122 = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 22 (C\u00e2u 16 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A (\u22122; 0; 0), B (0; 3; 0), C (0; 0; 4). M\u1eb7t ph\u1eb3ng (ABC) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x yz xyz xyz xy z A \u22122 + 3 + 4 = 1. B + + = 1. C 2 + \u22123 + 4 = 1. D 2 + 3 + \u22124 = 1. 234 \u0253 L\u1eddi gi\u1ea3i. x yz Ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (ABC) theo \u0111o\u1ea1n ch\u1eafn l\u00e0 \u22122 + 3 + 4 = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 505 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0104 C\u00e2u 23 (C\u00e2u 9 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A(\u22121; 0; 0), B(0; 2; 0) v\u00e0 C(0; 0; 3). M\u1eb7t ph\u1eb3ng (ABC) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 xyz xyz xy z 1 + \u22122 + 3 = 1. x yz + + = 1. A 1 + 2 + \u22123 = 1. B C \u22121 + 2 + 3 = 1. D 123 \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 24 (C\u00e2u 27 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng \u0111i qua O v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #n\u00bb = (1; \u22122; 5) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x + 2y \u2212 5z = 0. B x + 2y \u2212 5z + 1 = 0. C x \u2212 2y + 5z = 0. D x \u2212 2y + 5z + 1 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua O v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #n\u00bb = \u0253 L\u1eddi gi\u1ea3i. ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 (1; \u22122; 5) l\u00e0m v\u00e9c-t\u01a1 (x \u2212 0) \u2212 2(y \u2212 0) + 3(z \u2212 0) = 0 \u21d4 x \u2212 2y + 5z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 25 (C\u00e2u 13 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng \u0111i qua O v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #n\u00bb = (2; \u22121; 4) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 A 2x + y \u2212 4z + 1 = 0. B 2x + y \u2212 4z = 0. C 2x \u2212 y + 4z = 0. D 2x \u2212 y + 4z + 1 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua O v\u00e0 v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #n\u00bb \u0253 L\u1eddi gi\u1ea3i. l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 = (2; \u22121; 4) 2x \u2212 y + 4z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 26 (C\u00e2u 10 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng \u0111i qua O v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #n\u00bb = (1; 2; \u22123) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x + 2y \u2212 3z + 1 = 0. B x \u2212 2y + 3z + 1 = 0. C x \u2212 2y + 3z = 0. D x + 2y \u2212 3z = 0. M\u1eb7t ph\u1eb3ng \u0111i qua O(0; 0; 0) v\u00e0 nh\u1eadn #n\u00bb = \u0253 L\u1eddi gi\u1ea3i. v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x+ (1; 2; \u22123) l\u00e0m 2y \u2212 3z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 27 (C\u00e2u 1 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng \u0111i qua O v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #n\u00bb = (2; 3; \u22124) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x \u2212 3y + 4z + 1 = 0. B 2x + 3y \u2212 4z + 1 = 0. C 2x \u2212 3y + 4z = 0. D 2x + 3y \u2212 4z = 0. M\u1eb7t ph\u1eb3ng \u0111i qua O(0; 0; 0) v\u00e0 nh\u1eadn #n\u00bb \u0253 L\u1eddi gi\u1ea3i. ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh = (2; 3; \u22124) l\u00e0m v\u00e9c-t\u01a1 2x + 3y \u2212 4z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 506 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 28 (C\u00e2u 20 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (Oyz) l\u00e0 A z = 0. B x = 0. C x + y + z = 0. D y = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (Oyz) nh\u1eadn #i\u00bb = (1; 0; 0) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn v\u00e0 \u0111i qua g\u1ed1c t\u1ecda \u0111\u1ed9 O(0; 0; 0) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 29 (C\u00e2u 4 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (Oyz) l\u00e0 A x = 0. B x + y + z = 0. C z = 0. D y = 0. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (Oyz) l\u00e0 x = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 30 (C\u00e2u 10 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 A y = 0. B x = 0. C x + y = 0. D z = 0. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 31 (C\u00e2u 9 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y + z \u2212 5 = 0. \u0110i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c (P )? A Q(2; \u22121; 5). B P (0; 0; \u22125). C N (\u22125; 0; 0). D M (1; 1; 6). \u0253 L\u1eddi gi\u1ea3i. S\u1eed d\u1ee5ng ch\u1ee9c n\u0103ng CALC c\u1ee7a MTCT t\u00ecm \u0111\u01b0\u1ee3c M (1; 1; 6). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 32 (C\u00e2u 22 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i qua \u0111i\u1ec3m M (1; 2; \u22123) v\u00e0 c\u00f3 m\u1ed9t v\u00e9ct\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (1; \u22122; 3)? A x \u2212 2y + 3z \u2212 12 = 0. B x \u2212 2y \u2212 3z + 6 = 0. C x \u2212 2y + 3z + 12 = 0. D x \u2212 2y \u2212 3z \u2212 6 = 0. \u0253 L\u1eddi gi\u1ea3i. \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c A(x \u2212 x0) + B(y \u2212 y0) + C(z \u2212 z0) = 0 ta \u0111\u01b0\u1ee3c: (x \u2212 1) \u2212 2(y \u2212 2) + 3(z + 3) = 0 \u21d4 x \u2212 2y + 3z + 12 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 33 (C\u00e2u 43 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 3x \u2212 z + 2 = 0. Vect\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P )? C n#\u00bb3 = (3; \u22121; 0). D n#\u00bb2 = (3; 0; \u22121). A n#\u00bb4 = (\u22121; 0; \u22121). B n#\u00bb1 = (3; \u22121; 2). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 507 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 : (P ) : 3x + 0y \u2212 z + 2 = 0 n\u00ean (3; 0; \u22121) l\u00e0 t\u1ecda \u0111\u1ed9 vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P ). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 34 (C\u00e2u 15 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (\u03b1) : 3x + 2y \u2212 4z + 1 = 0. Vect\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (\u03b1)? A n#\u00bb2 = (3 ; 2 ; 4). B n#\u00bb3 = (2 ; \u22124 ; 1). C n#\u00bb1 = (3 ; \u22124 ; 1). D n#\u00bb4 = (3 ; 2 ; \u22124). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (\u03b1) : 3x + 2y \u2212 4z + 1 = 0 c\u00f3 m\u1ed9t vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 n#\u00bb4 (3 ; 2 ; \u22124). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 35 (C\u00e2u 47 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111i\u1ec3m A(0; 1; 1) v\u00e0 B(1; 2; 3). Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng AB. A x + y + 2z \u2212 3 = 0. B x + y + 2z \u2212 6 = 0. C x + 3y + 4z \u2212 7 = 0. D x + 3y + 4z \u2212 26 = 0. \u0253 L\u1eddi gi\u1ea3i. #\u00bb M\u1eb7t ph\u1eb3ng (P ) qua A v\u00e0 nh\u1eadn AB = (1; 1; 2) l\u00e0m vect\u01a1 ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x + (y \u2212 1) + 2(z \u2212 1) = 0 \u21d4 x + y + 2z \u2212 3 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 36 (C\u00e2u 45 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho 3 \u0111i\u1ec3m A(1; 0; 0); B(0; \u22122; 0);C(0; 0; 3). Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi d\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (ABC)? xyz xy z xyz x yz 1 + \u22122 + 3 = 1. 3 + 1 + \u22122 = 1. A 3 + \u22122 + 1 = 1. B \u22122 + 1 + 3 = 1. C D \u0253 L\u1eddi gi\u1ea3i. xyz Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng theo \u0111o\u1ea1n ch\u1eafn \u0111i qua 3 \u0111i\u1ec3m A, B, C l\u00e0 1 + \u22122 + 3 = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 37 (C\u00e2u 20 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng \u0111i qua \u0111i\u1ec3m A(2; \u22121; 2) v\u00e0 song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ) : 2x\u2212 y + 3z + 2 = 0 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x \u2212 y + 3z \u2212 9 = 0. B 2x \u2212 y + 3z + 11 = 0. C 2x \u2212 y \u2212 3z + 11 = 0. D 2x \u2212 y + 3z \u2212 11 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi m\u1eb7t ph\u1eb3ng (Q) song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ), m\u1eb7t ph\u1eb3ng (Q) c\u00f3 d\u1ea1ng 2x\u2212y +3z +D = 0 (D = 2). A(2; \u22121; 2) \u2208 (Q) \u21d2 D = \u221211. V\u1eady m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 2x \u2212 y + 3z \u2212 11 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 38 (C\u00e2u 17 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A(\u22121; 1; 1), B(2; 1; 0), C(1; \u22121; 2). M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 508 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng BC c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x + 2y \u2212 2z + 1 = 0. B x + 2y \u2212 2z \u2212 1 = 0. C 3x + 2z \u2212 1 = 0. D 3x + 2z + 1 = 0. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 B# C\u00bb ==\u2212(\u2212B# 1C\u00bb; \u2212=2;(21); l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) c\u1ea7n t\u00ecm. \u21d2 #n\u00bb 2; \u22122) c\u0169ng l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ). V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) l\u00e0 1(x + 1) + 2(y \u2212 1) \u2212 2(z \u2212 1) \u21d4 x + 2y \u2212 2z + 1 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 39 (C\u00e2u 23 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A (5; \u22124; 2) v\u00e0 B (1; 2; 4). M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x \u2212 3y \u2212 z + 8 = 0. B 3x \u2212 y + 3z \u2212 13 = 0. C 2x \u2212 3y \u2212 z \u2212 20 = 0. D 3x \u2212 y + 3z \u2212 25 = 0. \u0253 L\u1eddi gi\u1ea3i. C\u00f3 A# B\u00bb = (\u22124; 6; 2). Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng AB l\u00e0 \u22124(x \u2212 5) + 6(y + 4) + 2(z \u2212 2) = 0 \u21d4 2x \u2212 3y \u2212 z \u2212 20 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 40 (C\u00e2u 30 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(1; 3; 0) v\u00e0 B(5; 1; \u22121). M\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x \u2212 y \u2212 z + 5 = 0. B 2x \u2212 y \u2212 z \u2212 5 = 0. C x + y + 2z \u2212 3 = 0. D 3x + 2y \u2212 z \u2212 14 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB, do \u0111\u00f3 (P ) \u0111i qua trung \u0111i\u1ec3m I(3; 2; \u22121) c\u1ee7a AB, c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn #n\u00bbP = 1A# B\u00bb = (2; \u22121; \u22121). 2 Suy ra (P ) : 2(x \u2212 3) \u2212 1(y \u2212 2) \u2212 1(z + 1) = 0 \u21d4 2x \u2212 y \u2212 z \u2212 5 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 41 (C\u00e2u 27 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(\u22121; 2; 0) v\u00e0 B(3; 0; 2). M\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x + y + z \u2212 4 = 0. B 2x \u2212 y + z \u2212 2 = 0. C x + y + z \u2212 3 = 0. D 2x \u2212 y + z + 2 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi I Al#\u00e0B\u00bbtr=un(g4;\u0111\u2212i\u1ec32m; 2c)\u1ee7. a \u0111o\u1ea1n th\u1eb3ng AB. Suy ra I(1; 1; 1). Ta c\u00f3 #\u00bb M\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB \u0111i qua trung \u0111i\u1ec3m I c\u1ee7a AB v\u00e0 nh\u1eadn AB l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn, n\u00ean c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 (\u03b1) : 2x \u2212 y + z \u2212 2 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 509 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0104 C\u00e2u 42 (C\u00e2u 27 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(2; 1; 2) v\u00e0 B(6; 5; \u22124). M\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x + 2y \u2212 3z \u2212 17 = 0. B 4x + 3y \u2212 z \u2212 26 = 0. C 2x + 2y \u2212 3z + 17 = 0. D 2x + 2y + 3z \u2212 11 = 0. \u0253 L\u1eddi gi\u1ea3i. tMu\u1eb7yt\u1ebfnphl\u00e0\u1eb3nA#gB\u00bbtr=un(g4;tr4\u1ef1; c\u2212c6\u1ee7)an\u0111\u00eano\u1ea1cn\u00f3tphh\u1eb3\u01b0n\u01a1gnAgBtr\u0111\u00ecnihqul\u00e0a trung \u0111i\u1ec3m c\u1ee7a AB l\u00e0 M (4; 3; \u22121) v\u00e0 c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p 4(x \u2212 4) + 4(y \u2212 3) \u2212 6(z + 1) = 0 \u21d4 2(x \u2212 4) + 2(y \u2212 3) \u2212 3(z + 1) = 0 \u21d4 2x + 2y \u2212 3z \u2212 17 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 43 (C\u00e2u 19 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(4; 0; 1) v\u00e0 B(\u22122; 2; 3). M\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 6x \u2212 2y \u2212 2z \u2212 1 = 0. B 3x + y + z \u2212 6 = 0. C x + y + 2z \u2212 6 = 0. D 3x \u2212 y \u2212 z = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB c\u00f3 v\u00e9ct\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 A# B\u00bb = (\u2212 6; 2; 2) v\u00e0 \u0111i qua trung \u0111i\u1ec3m I(1; 1; 2) c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB. Do \u0111\u00f3, ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111\u00f3 l\u00e0 \u2212 6(x \u2212 1) + 2(y \u2212 1) + 2(z \u2212 2) = 0 \u21d4 \u22126x + 2y + 2z = 0 \u21d4 3x \u2212 y \u2212 z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 44 (C\u00e2u 34 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng \u0111i qua \u0111i\u1ec3m M (1; 1 \u2212 1) v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng x+1 y\u22122 z\u22121 \u2206 : = = c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 221 A 2x + 2y + z + 3 = 0. B x \u2212 2y \u2212 z = 0. C 2x + 2y + z \u2212 3 = 0. D x \u2212 2y \u2212 z \u2212 2 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (\u03b1)l\u00e0 m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm. x+1 y\u22122 z\u22121 V\u00ec (\u03b1) vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng \u2206 : = = n\u00ean (\u03b1)nh\u1eadn v\u00e9c t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng 2 2 1 #u\u00bb (2; 2; 1)c\u1ee7a \u2206l\u00e0 v\u00e9c t\u01a1 ph\u00e1p tuy\u1ebfn. L\u1ea1i c\u00f3, (\u03b1)\u0111i qua M (1; 1 \u2212 1). Do \u0111\u00f3, ph\u01b0\u01a1ng tr\u00ecnh (\u03b1)c\u00f3 d\u1ea1ng: 2 (x \u2212 1) + 2 (y \u2212 1) + (z + 1) = 0 \u21d4 2x + 2y + z \u2212 3 = 0 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 45 (C\u00e2u 37 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 510 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN x\u22123 y\u22121 z+1 Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; 1; 0) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u2206 : 1 = 4 = \u22122 . M\u1eb7t ph\u1eb3ng \u0111i qua M vu\u00f4ng g\u00f3c v\u1edbi \u2206 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 3x + y \u2212 z \u2212 7 = 0. B x + 4y \u2212 2z + 6 = 0. C x + 4y \u2212 2z \u2212 6 = 0. D 3x + y \u2212 z + 7 = 0. \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng \u2206 c\u00f3 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (1; 4; \u22122). G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm. Do \u2206 \u22a5 (P ) n\u00ean (P ) nh\u1eadn #u\u00bb l\u00e0m vect\u01a1 ph\u00e1p tuy\u1ebfn. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) l\u00e0: 1(x \u2212 2) + 4(y \u2212 1) + \u22122z = 0 \u21d4 x + 4y \u2212 2z \u2212 6 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 46 (C\u00e2u 23 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho 3 \u0111i\u1ec3m A (2; 0; 0), B (0; \u22121; 0), C (0; 0; 3). M\u1eb7t ph\u1eb3ng (ABC) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x yz xy z A + + = 1. B x2 + \u2212y1 + z\u22123 = 1. C \u2212x 2 y 1 z 3 D 2 + \u22121 + 3 = 1. + + = 1. 213 \u0253 L\u1eddi gi\u1ea3i. V\u1edbi 3 \u0111i\u1ec3m A (2; 0; 0), B (0; \u22121; 0), C (0; 0; 3), theo ph\u01b0\u01a1ng tr\u00ecnh \u0111o\u1ea1n ch\u1eafn ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t xyz ph\u1eb3ng (ABC) : 2 + \u22121 + 3 = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 47 (C\u00e2u 28 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (3; \u22122; 2), \u0111\u01b0\u1eddng th\u1eb3ng d: x\u22123 = y+1 = z\u22121 M\u1eb7t 1 2 \u22122 . ph\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x + 2y \u2212 2z + 5 = 0. B 3x \u2212 2y + 2z \u2212 17 = 0. C 3x \u2212 2y + 2z + 17 = 0. D x + 2y \u2212 2z \u2212 5 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (\u03b1) l\u00e0 m\u1eb7t ph\u1eb3ng \u0111i qua M (3; \u22122; 2) v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d: x \u2212 3 = y +1 = z\u22121 Vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d l\u00e0 #u\u00bb = (1; 2; \u22122). 1 2 \u22122 . (\u03b1) \u22a5 d n\u00ean vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (\u03b1) l\u00e0 #n\u00bb = (1; 2; \u22122). Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (\u03b1) l\u00e0 1 (x \u2212 3) + 2 (y + 2) \u2212 2 (z \u2212 2) = 0 \u21d4 x + 2y \u2212 2z + 5 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 48 (C\u00e2u 30 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian 0xyz cho \u0111i\u1ec3m M (2; 1; \u22122) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 3x \u2212 2y + z + 1 = 0. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t ph\u1eb3ng \u0111i qua M v\u00e0 song song v\u1edbi (P ) l\u00e0 A 2x + y \u2212 2z + 9 = 0. B 2x + y \u2212 2z \u2212 9 = 0. C 3x \u2212 2y + z + 2 = 0. D 3x \u2212 2y + z \u2212 2 = 0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 511","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng G\u1ecdi (Q) l\u00e0 m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm, ta c\u00f3 (Q) song song v\u1edbi (P ) n\u00ean ph\u01b0\u01a1ng tr\u00ecnh (Q) c\u00f3 d\u1ea1ng (Q) : 3x \u2212 2y + z + D = 0. M\u1eb7t kh\u00e1c, (Q) \u0111i qua M (2; 1; \u22122) n\u00ean ta c\u00f3 3 \u00b7 2 \u2212 2 \u00b7 1 + 1 \u00b7 (\u22122) + D = 0 \u21d2 D = \u22122. V\u1eady (Q) : 3x \u2212 2y + z \u2212 2 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 49 (C\u00e2u 34 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(1; 0; 0) v\u00e0 B(4; 1; 2). M\u1eb7t ph\u1eb3ng \u0111i qua A vu\u00f4ng g\u00f3c v\u1edbi AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 3x + y + 2z \u2212 17 = 0. B 3x + y + 2z \u2212 3 = 0. C 5x + y + 2z \u2212 5 = 0. D 5x + y + 2z \u2212 25 = 0. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 A# B\u00bb = (3; 1; 2) \u21d2 #n\u00bb(P ) = (3; 1; 2). Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi AB l\u00e0 3(x \u2212 1) + y + 2z = 0 \u21d4 3x + y + 2z \u2212 3 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 50 (C\u00e2u 30 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(0; 0; 1) v\u00e0 B(2; 1; 3). M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x + y + 2z \u2212 11 = 0. B 2x + y + 2z \u2212 2 = 0. C 2x + y + 4z \u2212 4 = 0. D 2x + y + 4z \u2212 17 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm. V\u00ec (P ) A\u22a5# B\u00bbA=B n\u00ean (P ) c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 A# B\u00bb = (2; 1; 2). M\u1eb7t ph\u1eb3ng (P ) qua A(0; 0; 1) v\u00e0 nh\u1eadn (2; 1; 2) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 2(x \u2212 0) + 1(y \u2212 0) + 2(z \u2212 1) = 0 \u21d4 2x + y + 2z \u2212 2 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 51 (C\u00e2u 35 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(0; 0; 1) v\u00e0 B(1; 2; 3). M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x + 2y + 2z \u2212 11 = 0. B x + 2y + 2z \u2212 2 = 0. C x + 2y + 4z \u2212 4 = 0. D x + 2y + 4z \u2212 17 = 0. \u0253 L\u1eddi gi\u1ea3i. V\u00ec m\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi AB n\u00ean nh\u1eadn A# B\u00bb = (1; 2; 2) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn, do \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 1(x \u2212 0) + 2(y \u2212 0) + 2(z \u2212 1) = 0 \u21d4 x + 2y + 2z \u2212 2 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 512 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 52 (C\u00e2u 38 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(1; 0; 0) v\u00e0 B(3; 2; 1). M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi AB c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x + 2y + z \u2212 2 = 0. B 3x + 2y + z \u2212 17 = 0. C 4x + 2y + z \u2212 4 = 0. D 2x + 2y + z \u2212 11 = 0. \u0253 L\u1eddi gi\u1ea3i. #\u00bb M\u1eb7t ph\u1eb3ng \u0111i qua A(1; 0; 0) v\u00e0 nh\u1eadn AB = (2; 2; 1) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng c\u00f3 d\u1ea1ng 2(x \u2212 1) + 2(y \u2212 0) + (z \u2212 0) = 0 \u21d4 2x + 2y + z \u2212 2 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 53 (C\u00e2u 30 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; \u22121) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x + y \u2212 3z + 1 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x + y \u2212 3z \u2212 7 = 0. B 2x + y \u2212 3z + 7 = 0. C 2x + y + 3z \u2212 1 = 0. D 2x + y + 3z + 1 = 0. \u0253 L\u1eddi gi\u1ea3i. V\u00ec m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ) n\u00ean c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (2; 1; \u22123). Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ) l\u00e0 2(x \u2212 1) + (y \u2212 2) \u2212 3(z + 1) = 0 \u21d4 2x + y \u2212 3z \u2212 7 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 54 (C\u00e2u 36 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A (1; \u22121; 2) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x + 2y \u2212 3z + 1 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x + 2y + 3z \u2212 5 = 0. B x + 2y + 3z + 5 = 0. C x + 2y \u2212 3z \u2212 7 = 0. D x + 2y \u2212 3z + 7 = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng song song v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh d\u1ea1ng x + 2y \u2212 3z + C = 0 (C = 1). V\u00ec m\u1eb7t ph\u1eb3ng n\u00e0y \u0111i qua A n\u00ean ta c\u00f3 1 \u2212 2 \u00b7 1 \u2212 3 \u00b7 2 + C = 0 \u21d2 C = 7. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 x + 2y \u2212 3z + 7 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 55 (C\u00e2u 37 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; \u22121) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y + 3z + 1 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x + 2y + 3z + 2 = 0. B x \u2212 2y + 3z \u2212 6 = 0. C x \u2212 2y + 3z + 6 = 0. D x + 2y + 3z \u2212 2 = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 (x \u2212 1) \u2212 2(y \u2212 2) + 3(z + 1) = 0 \u21d4 x \u2212 2y + 3z + 6 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C 513 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0104 C\u00e2u 56 (C\u00e2u 37 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(0; \u22123; 2) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 y + 3z + 5 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x \u2212 y + 3z + 9 = 0. B 2x + y + 3z \u2212 3 = 0. C 2x + y + 3z + 3 = 0. D 2x \u2212 y + 3z \u2212 9 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (Q) l\u00e0 m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm. Theo gi\u1ea3 thi\u1ebft (Q) \u2225 (P ) n\u00ean (Q) : 2x \u2212 y + 3z + m = 0 (m = 5). M\u00e0 (Q) qua A n\u00ean 2.0 \u2212 (\u22123) + 3.2 + m = 0 \u21d4 m = \u22129. V\u1eady m\u1eb7t ph\u1eb3ng (Q) : 2x \u2212 y + 3z \u2212 9 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 57 (C\u00e2u 32 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(0; \u22123; 2) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 y + 3z + 5 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 v\u00e0 song song v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x \u2212 y + 3z + 9 = 0. B 2x + y + 3z \u2212 3 = 0. C 2x + y + 3z + 3 = 0. D 2x \u2212 y + 3z \u2212 9 = 0. \u0253 L\u1eddi gi\u1ea3i. V\u00ec m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ) n\u00ean m\u1eb7t ph\u1eb3ng n\u00e0y nh\u1eadn v\u00e9c-t\u01a1 #n\u00bbP = (2; \u22121; 3) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn v\u00e0 do m\u1eb7t ph\u1eb3ng qua A(1; 2; \u22121) n\u00ean c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 2(x \u2212 0) \u2212 (y + 3) + 3(z \u2212 2) = 0 \u21d4 2x \u2212 y + 3z \u2212 9 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 58 (C\u00e2u 37 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; \u22122; 1) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 3y \u2212 z + 1 = 0. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 2 + 2t \uf8f1x = 2 + 2t \uf8f1x = 2 + 2t \uf8f1x = 2 + 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22122 + 3t . B y = 2 \u2212 3t . C y = \u22122 \u2212 3t . D y = \u22123 \u2212 2t . \uf8f4\uf8f3z = 1 + t \uf8f4\uf8f3z = 1 \u2212 t \uf8f4\uf8f3z = 1 \u2212 t \uf8f3\uf8f4z = \u22121 + t \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (P ) c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = #n\u00bb(P) = (2; \u22123; \u22121). \uf8f1x = 2 + 2t \uf8f4 \uf8f2 \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 y = \u22122 \u2212 3t \uf8f4\uf8f3z = 1 \u2212 t. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 59 (C\u00e2u 45 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 3x + 4y + 2z + 4 = 0 v\u00e0 \u0111i\u1ec3m A(1; \u22122; 3). T\u00ednh kho\u1ea3ng c\u00e1ch d t\u1eeb A \u0111\u1ebfn (P ). \u221a A d= 5 B d= 5 C d = \u221a5 . D d= 5 . . 29 . 9 29 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 d(A; (P )) = |3.1 +\u221a4.(\u22122) + 2.3 + 4| = \u221a5 32 + 42 + 22 29 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 514 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 60 (C\u00e2u 42 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 6x \u2212 2y + z \u2212 35 = 0 v\u00e0 \u0111i\u1ec3m A(\u22121; 3; 6). G\u1ecd\u221ai A l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edb\u221ai A qua (P ). T\u00ednh OA\u221a. \u221a A OA = 3 26. B OA = 5 3. C OA = 46. D OA = 186. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi d l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi mp (P ) n\u00ean d c\u00f3 VTCP l\u00e0 u#\u00bbd = n# P\u00bb = (6; \u22122; 1) \uf8f1x = \u22121 + 6t \uf8f4 \uf8f2 PTTS c\u1ee7a d : y = 3 \u2212 2t \uf8f3\uf8f4z = 6 + t. G\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a A tr\u00ean mp (P ). Khi \u0111\u00f3 t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m H l\u00e0 nghi\u1ec7m h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh: \uf8f1x = \u22121 + 6t \uf8f1t = 1 \uf8f4\uf8f4 \uf8f4\uf8f4 \uf8f2\uf8f4y = 3 \u2212 2t \uf8f4\uf8f2x = 5 \u21d4 Suy ra H(5; 1; 7). \uf8f4z = 6 + t \uf8f4y = 1 \uf8f4\uf8f4 \uf8f4 \u2212 2y + z \u2212 35 = 0 \uf8f4 \uf8f36x \uf8f3z = 7 V\u00ec A l\u00e0 \u0111i\u1ec3m\u221a \u0111\u1ed1i x\u1ee9ng c\u1ee7a A qua (P ) n\u00ean H l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AA . Suy ra A (11; \u22121; 8). V\u1eady OA = 186. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 61 (C\u00e2u 14 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz kho\u1ea3ng c\u00e1ch gi\u1eefa hai m\u1eb7t ph\u1eb3ng (P ) : x + 2y + 2z \u2212 10 = 0 v\u00e0 (Q) : x + 2y + 2z \u2212 3 = 0 b\u1eb1ng A 8 B 7 C 3. D 4 . . . 3 3 3 \u0253 L\u1eddi gi\u1ea3i. X\u00e9t th\u1ea5y (P ) \u2225 (Q). Tr\u00ean (P ) l\u1ea5y M (0; 0; 5). Khi \u0111\u00f3, kho\u1ea3ng c\u00e1ch gi\u1eefa hai m\u1eb7t ph\u1eb3ng (P ) v\u00e0 (Q) l\u00e0: d ((P ), (Q)) = d (M, (Q)) = |0 +\u221a2 \u00b7 0 + 2 \u00b7 5 \u2212 3| = 7 . 12 + 22 + 22 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 62 (C\u00e2u 29 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(3; 2; \u22121) v\u00e0 \u0111i qua \u0111i\u1ec3m A(2; 1; 2). M\u1eb7t ph\u1eb3ng n\u00e0o d\u01b0\u1edbi \u0111\u00e2y ti\u1ebfp x\u00fac v\u1edbi (S) t\u1ea1i A? A x + y \u2212 3z \u2212 8 = 0. B x \u2212 y \u2212 3z + 3 = 0. C x + y + 3z \u2212 9 = 0. D x + y \u2212 3z + 3 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (P ) l\u00e0 mI#A\u1eb7\u00bbt=ph(\u1eb3\u2212n1g; c\u1ea7n t\u00ecm. Khi \u0111\u00f3 (P ) ti\u1ebfp x\u00fac v\u1edbi (S) t\u1ea1i A khi ch\u1ec9 khi (P ) \u0111i qua A(2; 1; 2) v\u00e0 nh\u1eadn vect\u01a1 \u22121; 3) l\u00e0m vect\u01a1 ph\u00e1p tuy\u1ebfn. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) l\u00e0 \u2212x \u2212 y + 3z \u2212 3 = 0 \u21d4 x + y \u2212 3z + 3 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 63 (C\u00e2u 37 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; \u22121; 4) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 3x \u2212 2y + z + 1 = 0. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i qua M v\u00e0 song song v\u1edbi (P ) l\u00e0 A 2x \u2212 y + 4z \u2212 21 = 0. B 2x \u2212 y + 4z + 21 = 0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 515 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng C 3x \u2212 2y + z \u2212 12 = 0. D 3x \u2212 2y + z + 12 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (\u03b1) l\u00e0 m\u1eb7t ph\u1eb3ng \u0111i qua M v\u00e0 song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ). Do m\u1eb7t ph\u1eb3ng (\u03b1) song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ) n\u00ean (\u03b1) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (3; \u22122; 1). M\u1eb7t ph\u1eb3ng (\u03b1) \u0111i qua \u0111i\u1ec3m M (2; \u22121; 4), v\u1eady (\u03b1) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 3 \u00b7 (x \u2212 2) \u2212 2 \u00b7 (y + 1) + 1 \u00b7 (z \u2212 4) = 0 \u21d4 3x \u2212 2y + z \u2212 12 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 64 (C\u00e2u 31 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; \u22121; 3) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 3x \u2212 2y + z + 1 = 0. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i qua M v\u00e0 song song v\u1edbi (P ) l\u00e0 A 3x \u2212 2y + z + 11 = 0. B 2x \u2212 y + 3z \u2212 14 = 0. C 3x \u2212 2y + z \u2212 11 = 0. D 2x \u2212 y + 3z + 14 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (\u03b1) l\u00e0 m\u1eb7t ph\u1eb3ng \u0111i qua M v\u00e0 song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ). Do m\u1eb7t ph\u1eb3ng (\u03b1) song song v\u1edbi m\u1eb7t ph\u1eb3ng (P ) n\u00ean (\u03b1) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (3; \u22122; 1). M\u1eb7t ph\u1eb3ng (\u03b1) \u0111i qua \u0111i\u1ec3m M (2; \u22121; 3), v\u1eady (\u03b1) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 3 \u00b7 (x \u2212 2) \u2212 2 \u00b7 (y + 1) + 1 \u00b7 (z \u2212 3) = 0 \u21d4 3x \u2212 2y + z \u2212 11 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 65 (C\u00e2u 30 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; 1; \u22123) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 3x \u2212 2y + z \u2212 3 = 0. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i qua M v\u00e0 song song v\u1edbi (P ) l\u00e0 A 3x \u2212 2y + z + 1 = 0. B 3x \u2212 2y + z \u2212 1 = 0. C 2x + y \u2212 3z + 14 = 0. D 2x + y \u2212 3z \u2212 14 = 0. \u0253 L\u1eddi gi\u1ea3i. Ta th\u1ea5y 3 \u00b7 2 \u2212 2 \u00b7 1 \u2212 3 \u2212 3 = \u22122 = 0 n\u00ean M \u2208\/ (P ). V\u00ec v\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng qua M v\u00e0 song song v\u1edbi (P ) l\u00e0 3(x \u2212 2) \u2212 2(y \u2212 1) + (z + 3) = 0 \u21d4 3x \u2212 2y + z \u2212 1 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 66 (C\u00e2u 38 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; \u22121; 2) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 y + 3z + 1 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x + y + 3z + 7 = 0. B 2x + y + 3z \u2212 7 = 0. C 2x \u2212 y + 3z + 9 = 0. D 2x \u2212 y + 3z \u2212 9 = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (Q) l\u00e0 m\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi (P ). V\u00ec (Q) \u2225 (P ) n\u00ean ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t ph\u1eb3ng (Q) c\u00f3 d\u1ea1ng 2x \u2212 y + 3z + m = 0, v\u1edbi m = 1. M\u1eb7t kh\u00e1c, m\u1eb7t ph\u1eb3ng (Q) \u0111i qua A n\u00ean 2 \u00b7 1 \u2212 (\u22121) + 3 \u00b7 2 + m = 0 \u21d4 m = \u22129 (th\u1ecfa m\u00e3n). V\u1eady, m\u1eb7t ph\u1eb3ng (Q) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh 2x \u2212 y + 3z \u2212 9 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 516 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 67 (C\u00e2u 33 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; 3). Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t c\u1ea7u t\u00e2m A v\u00e0 ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng x \u2212 2y + 2x + 3 = 0 l\u00e0 A (x + 1)2 + (y + 2)2 + (z + 3)2 = 2. B (x \u2212 1)2 + (y \u2212 2)2 + (z \u2212 3)2 = 2. C (x + 1)2 + (y + 2)2 + (z + 3)2 = 4. D (x \u2212 1)2 + (y \u2212 2)2 + (z \u2212 3)2 = 4. \u0253 L\u1eddi gi\u1ea3i. B\u00e1n k\u00ednh m\u1eb7t c\u1ea7u l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m A \u0111\u1ebfn m\u1eb7t ph\u1eb3ng \u0111\u00e3 cho n\u00ean |1 \u2212 2.2 + 2.3 + 3| 6 R = = = 2. 1 + (\u22122)2 + 22 3 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t c\u1ea7u l\u00e0 4(x \u2212 1)2 + (y \u2212 2)2 + (z \u2212 3)2 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 68 (C\u00e2u 19 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i x\u22121 \u22123 qua \u0111i\u1ec3m M (3; \u22121; 1) v\u00e0 vu\u00f4ng g\u00f3c \u0111\u01b0\u1eddng th\u1eb3ng \u2206 : 3 = y+2 = z ? \u22122 1 A 3x \u2212 2y + z + 12 = 0. B 3x + 2y + z \u2212 8 = 0. C 3x \u2212 2y + z \u2212 12 = 0. D x \u2212 2y + 3z + 3 = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi \u2206 nh\u1eadn u# \u2206\u00bb = (3; \u22122; 1) l\u00e0m vtpt \u21d2 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm c\u00f3 d\u1ea1ng 3(x \u2212 3) \u2212 2(y + 1) + (z \u2212 1) = 0 \u21d4 3x \u2212 2y + z \u2212 12 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 69 (C\u00e2u 10 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t ph\u1eb3ng (Oyz)? A y = 0. B x = 0. C y \u2212 z = 0. D z = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (Oyz) vu\u00f4ng g\u00f3c v\u1edbi tr\u1ee5c Ox do \u0111\u00f3 n\u00f3 nh\u1eadn (1, 0, 0) l\u00e0 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn, h\u01a1n n\u1eefa (Oyz) \u0111i qua \u0111i\u1ec3m O(0, 0, 0). V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (Oyz) l\u00e0 1(x \u2212 0) + 0(y \u2212 0) + 0(z \u2212 0) = 0 hay x = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 70 (C\u00e2u 26 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111i\u1ec3m A(4; 0; 1) v\u00e0 B(\u22122; 2; 3). Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB? A 3x \u2212 y \u2212 z = 0. B 3x + y + z \u2212 6 = 0. C 3x \u2212 y \u2212 z + 1 = 0. D 6x \u2212 2y \u2212 2z \u2212 1 = 0. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 A# B\u00bb(\u22126; 2; 2), trung \u0111i\u1ec3m c\u1ee7a AB l\u00e0 I(1; 1; 2). M\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a AB nh\u1eadn v\u00e9c-t\u01a1 #n\u00bb(3; \u22121; \u22121) l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn v\u00e0 \u0111i qua \u0111i\u1ec3m I(1; 1; 2). V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a AB l\u00e0 3(x \u2212 1) \u2212 (y \u2212 1) \u2212 (z \u2212 2) = 0 \u21d4 3x \u2212 y \u2212 z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 517 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0104 C\u00e2u 71 (C\u00e2u 33 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y \u2212 1)2 + (z + 2)2 = 2 v\u00e0 hai x\u22122 y z\u22121 x y z\u22121 \u0111\u01b0\u1eddng th\u1eb3ng d : == \u22121 , \u2206 : 1 = 1 = \u22121 . Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng 1 2 tr\u00ecnh c\u1ee7a m\u1ed9t m\u1eb7t ph\u1eb3ng ti\u1ebfp x\u00fac v\u1edbi (S), song song v\u1edbi d v\u00e0 \u2206? A x + z + 1 = 0. B x + y + 1 = 0. C y + z + 3 = 0. D x + z \u2212 1 = 0. \u221a \u0253 L\u1eddi gi\u1ea3i. (S) c\u00f3 t\u00e2m I(\u22121; 1; \u22122) v\u00e0 b\u00e1n k\u00ednh R = 2. u#\u00bb1(1; ph\u01b0\u01a1ng u#\u00bb2(1; 1; \u22121). d c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng \u22121). 2; \u22121), \u2206 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 t\u00ecm song song v\u1edbi d v\u00e0 \u2206 n\u00ean n\u00f3 nh\u1eadn #n\u00bb(1; 0; 1) Ta c\u00f3 [u#\u00bb1, u#\u00bb2] = (\u22121; 0; V\u00ec m\u1eb7t ph\u1eb3ng (P ) c\u1ea7n l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng. Ph\u01b0\u01a1ng tr\u00ecnh (P ) c\u00f3 d\u1ea1ng x + z + d = 0. V\u00ec (S) ti\u1ebfp x\u00fac v\u1edbi (P ) n\u00ean d(I, (P )) = R \u21d4 |d\u221a\u2212 3| = \u221a \u21d4 \u00f1d = 5 2 2 d=1 V\u1eady ta \u0111\u01b0\u1ee3c hai m\u1eb7t ph\u1eb3ng l\u00e0 x + z + 1 = 0 v\u00e0 x + z + 5 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 72 (C\u00e2u 2 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t ph\u1eb3ng (\u03b1) : x + y + z \u2212 6 = 0. \u0110i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y kh\u00f4ngthu\u1ed9c (\u03b1)? A N (2; 2; 2). B Q(3; 3; 0). C P (1; 2; 3). D M (1; \u22121; 1). \u0253 L\u1eddi gi\u1ea3i. Thay t\u1ecda \u0111\u1ed9 c\u1ee7a M v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (\u03b1) ta \u0111\u01b0\u1ee3c 1 \u2212 1 + 1 \u2212 6 = \u22125 = 0 \u21d2 M \u2208\/ (\u03b1). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 73 (C\u00e2u 15 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz cho ba \u0111i\u1ec3m M (2; 0; 0), N (0; \u22121; 0) v\u00e0 P (0; 0; 2). M\u1eb7t ph\u1eb3ng (M N P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 z xyz x y z2 A + + = 0. B x2 + \u2212y1 + = \u22121. C x2 \u2212y 1 z 2 D + + = 1. 2 + \u22121 + 2 = 1. 212 \u0253 L\u1eddi gi\u1ea3i. xyz S\u1eed d\u1ee5ng ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng theo \u0111o\u1ea1n ch\u1eafn ta \u0111\u01b0\u1ee3c: (M N P ) : 2 + \u22121 + 2 = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 74 (C\u00e2u 46 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). x \u2212 10 y \u2212 2 Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng \u2206 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh == 51 z+2 . 1 X\u00e9t m\u1eb7t ph\u1eb3ng (P ) : 10x + 2y + mz + 11 = 0, m l\u00e0 tham s\u1ed1 th\u1ef1c. T\u00ecm t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb c\u1ee7a m \u0111\u1ec3 m\u1eb7t ph\u1eb3ng (P ) vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng \u2206. A m = \u22122. B m = 2. C m = \u221252. D m = 52. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 518","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN Vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u2206 l\u00e0 u# \u2206\u00bb = (5; 1; 1). Vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P ) l\u00e0 #n\u00bb = (10; 2; m). \u2206 vu\u00f4ng g\u00f3c v\u1edbi (P ) khi v\u00e0 ch\u1ec9 khi u# \u2206\u00bb v\u00e0 #n\u00bb c\u00f9ng ph\u01b0\u01a1ng. Hay 10 = 2 = m suy ra m = 2. 5 1 1 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 75 (C\u00e2u 48 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111i\u1ec3m A(\u22122; 3; 1) v\u00e0 B(5; 6; 2). \u0110\u01b0\u1eddng th\u1eb3ng AB c\u1eaft m\u1eb7t ph\u1eb3ng (Oxz) t\u1ea1i \u0111i\u1ec3m M . T\u00ednh t\u1ec9 s\u1ed1 AM \u00b7 BM AM 1 AM AM 1 AM A =. B = 2. C =. D = 3. BM 2 BM BM 3 BM \u0253 L\u1eddi gi\u1ea3i. M \u2208 (Oxz) \u21d2 M (x; 0; z) ; #\u00bb (7; 3; 1) \u21d2 AB = \u221a ; #\u00bb = (x + 2; \u22123; z \u2212 1) v\u00e0 A, B, M th\u1eb3ng AB = 59 AM \uf8f1x + 2 = 7k \uf8f1x = \u22129 A# M\u00bb k.A# B\u00bb(k \uf8f4 \uf8f4 B# M\u00bb \uf8f2 \uf8f2 h\u00e0ng \u21d2 = \u2208 R) \u21d4 \u2212 3 = 3k \u21d4 \u22121 = k \u21d2 M (\u22129; 0; 0). = (\u221214; \u22126; \u22122) \u21d2 \u221a \uf8f3\uf8f4z \u2212 1 = k \uf8f3\uf8f4z = 0 BM = 118 = 2AB. C\u00e1ch kh\u00e1c AM = d(A; (Oxz)) = 1\u00b7 BM d(B; (Oxz)) 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 76 (C\u00e2u 47 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y + 2z \u2212 3 = 0 v\u00e0 m\u1eb7t c\u1ea7u (S) : x2 + y2 + z2 + 2x \u2212 4y \u2212 2z + 5 = 0. Gi\u1ea3 s\u1eed \u0111i\u1ec3m M \u2208 (P ) v\u00e0 N \u2208 (S) sao cho c\u00f9ng ph\u01b0\u01a1ng v\u1edbi #u\u00bb = (1; 0; 1) v\u00e0 kho\u1ea3ng c\u00e1ch gi\u1eefa M v\u00e0\u221aN l\u00e0 l\u1edbn nh\u1ea5t. T\u00ednh\u221aM N . A M N = 3. B M N = 1 + 2 2. C M N = 3 2. D M N = 14. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(\u22121; 2; 1) b\u00e1n k\u00ednh R = 1. Ta c\u00f3 d (I, (P )) = | \u2212 1 \u2212 4 + 2 \u2212 3| = 2 > R n\u00ean (P ) kh\u00f4ng c\u1eaft (S). 12 + (\u22122)2 + 22 G\u1ecdi d l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng qua I v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ). G\u1ecdi T l\u00e0 giao \u0111i\u1ec3m c\u1ee7a d v\u00e0 m\u1eb7t c\u1ea7u (S) th\u1ecfa d (T ; (P )) > d (I; (P )). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 519 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng N T I P Mo M HH Ta c\u00f3 d (T, (P )) = d (I, (P )) + R = 2 + 1 = 3. Ta c\u00f3 cos #u\u00bb, n# (P\u00bb) = 1.1 \u2212 2.0 \u221a+ 1.2 = \u221a1 . 1 + (\u22122)2 + 22. 12 + 02 + 12 2 \u0110\u01b0\u1eddng th\u1eb3ng M N c\u00f3 v\u00e9ct\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb n\u00ean ta c\u00f3 sin (M N, (P )) = |cos ( #u\u00bb, n# P\u00bb)| = \u221a1 \u21d2 (M N, (P )) = 45\u25e6. 2 NH \u221a G\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a N l\u00ean (P ). Ta c\u00f3 M N = sin 45\u25e6 = N H. 2. Do \u0111\u00f3 M N l\u1edbn nh\u1ea5t khi N H l\u1edbn nh\u1ea5t. \u0110i\u1ec1u n\u00e0y x\u1ea3y ra khi N \u2261 T v\u00e0 H \u2261 H v\u1edbi H l\u00e0\u221ah\u00ecnh ch\u221ai\u1ebfu c\u1ee7a I l\u00ean (P ). Khi \u0111\u00f3 N Hmax = T H = 3 v\u00e0 M Nmax = N Hmax. 2 = 3 2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 77 (C\u00e2u 42 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; \u22122). G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng ch\u1ee9a tr\u1ee5c Ox sao cho kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn (P ) l\u1edbn nh\u1ea5t. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (P ) l\u00e0 A 2y + z = 0. B 2y \u2212 z = 0. C y + z = 0. D y \u2212 z = 0. \u0253 L\u1eddi gi\u1ea3i. GTa\u1ecdicK\u00f3 Kl\u00e0(1h;\u00ec0n;h0)c,hA#i\u1ebfKu\u00bb vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(1; 2; \u22122) l\u00ean tr\u1ee5c Ox. A = (0; \u22122; 2). HK G\u1ecdi H l\u00e0 \u0111i\u1ec3m chi\u1ebfu c\u1ee7a A l\u00ean m\u1eb7\u221at ph\u1eb3ng (P ). Ta c\u00f3 d(A, (P )) = AH \u2264 \u221aAK = 2 2. Suy ra max d(A, (P )) = 2 2, \u0111\u1ea1t \u0111\u01b0\u1ee3c khi H \u2261 K(1; 0; 0). AK# Kh\u00bbi \u0111\u00f3 m\u1eb7t ph\u1eb3ng (P ) qua O(0; 0; 0) c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 = (0; \u22122; 2). N\u00ean ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) l\u00e0 0.(x \u2212 1) \u2212 2(y \u2212 0) + 2(z \u2212 0) = 0 \u21d4 y \u2212 z = 0. V\u1eady (P ) : y \u2212 z = 0. 520 S\u0110T: 0905.193.688 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 78 (C\u00e2u 46 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + (y \u2212 2)2 + (z \u2212 3)2 = 1 v\u00e0 \u0111i\u1ec3m A(2; 3; 4). X\u00e9t c\u00e1c \u0111i\u1ec3m M thu\u1ed9c (S) sao cho \u0111\u01b0\u1eddng th\u1eb3ng AM ti\u1ebfp x\u00fac v\u1edbi (S), M lu\u00f4n thu\u1ed9c m\u1eb7t ph\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x + 2y + 2z \u2212 15 = 0. B x + y + z \u2212 7 = 0. C 2x + 2y + 2z + 15 = 0. D x + y + z + 7 = 0. \u0253 L\u1eddi gi\u1ea3i. T\u1eeb gi\u1ea3 thi\u1ebft ta c\u00f3 I(1; 2; 3) l\u00e0 t\u00e2m c\u1ee7a m\u1eb7t c\u1ea7u (S); \u0111i\u1ec3m A(2; 3; 4) n\u1eb1m ngo\u00e0i (S). Do IA ti\u1ebfp x\u00fac v\u1edbi (S) t\u1ea1i M #n\u00ea\u00bbn I M \u22a5 AM . \u2212 2; z0 \u2212 3); #\u00bb = (x0 \u2212 2; y0 \u2212 3; z0 \u2212 4). L\u1ea5y M (x0; y0; z0) \u2208 (S) ta c\u00f3 IM = (x0 \u2212 1; y0 AM \u00aeM \u2208 (S) n\u00ean \u00ae(x0 \u2212 1)2 + (y0 \u2212 2)2 + (z0 \u2212 3)2 = 1 Do IM \u22a5 AM (x0 \u2212 1)(x0 \u2212 2) + (y0 \u2212 2)(y0 \u2212 3) + (z0 \u2212 3)(z0 \u2212 4) = 0 \u21d4 \u00ae(x0 \u2212 1)2 + (y0 \u2212 2)2 + (z0 \u2212 3)2 = 1 \u2212 2) + (z0 \u2212 3)2 \u2212 (z0 \u2212 3) = 0. (\u2217) (x0 \u2212 1)2 \u2212 (x0 \u2212 1) + (y0 \u2212 2)2 \u2212 (y0 T\u1eeb (\u2217) ta c\u00f3 x0 \u2212 1 + y0 \u2212 2 + z0 \u2212 3 = 1 \u21d4 x0 + y0 + z0 \u2212 7 = 0. V\u1eady M \u2208 (P ) : x + y + z \u2212 7 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 79 (C\u00e2u 33 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A (2; \u22122; 4), B (\u22123; 3; \u22121) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 y + 2z \u2212 8 = 0. X\u00e9t M l\u00e0 \u0111i\u1ec3m thay \u0111\u1ed5i thu\u1ed9c (P ), gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a 2M A2 + 3M B2 b\u1eb1ng A 135. B 105. C 108. D 145. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi I l\u00e0 \u0111i\u1ec3m th\u1ecfa m\u00e3n \u0111\u1eb3ng th\u1ee9c 2I#A\u00bb + 3I#B\u00bb = #0\u00bb. \uf8f1\uf8f42(xI \u2212 2) + 3(xI + 3) = 0 \uf8f1\uf8f45x1 + 5 = 0 \uf8f1\uf8f4x1 = \u22121 \uf8f2 \uf8f2\uf8f2 \u21d2 2(yI + 2) + 3(yI \u2212 3) = 0 \u21d4 5y1 \u2212 5 = 0 \u21d4 y1 = 1 \u21d2 I(\u22121; 1; 1). \uf8f3\uf8f42(zI \u2212 4) + 3(zI + 1) = 0 \uf8f3\uf8f45z1 \u2212 5 = 0 \uf8f3\uf8f4z1 = 1 Khi \u0111\u00f3 2M A2 + 3M B2 = 2M# A\u00bb2 + 3M# B\u00bb2 = 2(M# \u00bbI + I#A\u00bb)2 + 3(M# \u00bbI + I#B\u00bb)2 = 5M# \u00bbI2 + 2M# \u00bbI \u00b7 (2I#A\u00bb + 3I#B\u00bb) + 2I#A\u00bb2 + 3I#B\u00bb2 = 5M I2 + 2IA2 + 3IB2. V\u00ec A, B, I c\u1ed1 \u0111\u1ecbnh n\u00ean 2M A2 + 3M B2 nh\u1ecf nh\u1ea5t khi M I nh\u1ecf nh\u1ea5t hay M l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a \u0111i\u1ec3m I tr\u00ean m\u1eb7t ph\u1eb3ng (P ). \u21d2 \u2203k \u2208 R, I#M\u00bb = k #n\u00bb(P ) \u21d2 \uf8f4\uf8f1xM = 2k \u2212 1 \uf8f2 = \u2212k + 1 yM \uf8f4\uf8f3zM = 2k + 1. M\u00e0 M \u2208 (P ) \u21d2 2(2k \u2212 1) \u2212 (\u2212k + 1) + 2(2k + 1) \u2212 8 = 0 \u21d4 9k \u2212 9 = 0 \u21d4 k = 1 \u21d2 M (1; 0; 3). V\u1eady gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a 2M A2 + 3M B2 = 5M I2 + 2IA2 + 3IB2 = 135. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 80 (C\u00e2u 47 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(2; 1; \u22121). G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng ch\u1ee9a tr\u1ee5c Oy sao cho kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn (P ) l\u00e0 l\u1edbn nh\u1ea5t. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (P ) l\u00e0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 521 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng A 2x \u2212 z = 0. B 2x + z = 0. C x \u2212 z = 0. D x + z = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A l\u00ean (P ) v\u00e0 A l\u00e0 h\u00ecnh chi\u1ebfu A vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A l\u00ean tr\u1ee5c Oy \u21d2 A (0; 1; 0). Khi \u0111\u00f3 kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn (P ) l\u00e0 \u0111o\u1ea1n th\u1eb3ng AH \u2264 AA . \u0110\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng# A\u00bbH d\u00e0i nh\u1ea5t khi H v\u00e0 A tr\u00f9ng nhau. H Khi \u0111\u00f3 (P ) nh\u1eadn A A = (2; 0; \u22121) l\u00e0m#v\u00e9\u00bbct\u01a1 ph\u00e1p tuy\u1ebfn. A Suy ra (2; 0; \u22121) l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) : A A = 2(x \u2212 0) + 0(y \u2212 1) + (\u22121)(z \u2212 0) = 0 \u21d4 2x \u2212 z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 81 (C\u00e2u 20 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m M (3; \u22121; \u22122) v\u00e0 m\u1eb7t ph\u1eb3ng (\u03b1) : 3x \u2212 y + 2z + 4 = 0. Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111i qua M v\u00e0 song song v\u1edbi (\u03b1) A 3x + y \u2212 2z \u2212 14 = 0. B 3x \u2212 y + 2z + 6 = 0. C 3x \u2212 y + 2z \u2212 6 = 0. D 3x \u2212 y \u2212 2z + 6 = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng song song v\u1edbi (\u03b1) v\u00e0 qua \u0111i\u1ec3m M l\u00e0 3x \u2212 y + 2z \u2212 6 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 82 (C\u00e2u 38 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u \u0111i qua ba \u0111i\u1ec3m M (2; 3; 3), N (2; \u22121; \u22121), P (\u22122; \u22121; 3) v\u00e0 c\u00f3 t\u00e2m thu\u1ed9c m\u1eb7t ph\u1eb3ng (\u03b1) : 2x + 3y \u2212 z + 2 = 0? B x2 + y2 + z2 \u2212 4x + 2y \u2212 6z \u2212 2 = 0. A x2 + y2 + z2 \u2212 2x + 2y \u2212 2z \u2212 10 = 0. C x2 + y2 + z2 + 4x \u2212 2y + 6z + 2 = 0. D x2 + y2 + z2 \u2212 2x + 2y \u2212 2z \u2212 2 = 0. \u0253 L\u1eddi gi\u1ea3i. I(2; \u22121; 3) \u2208 (\u03b1); IM = IN = IP = 4. V\u1eady m\u1eb7t c\u1ea7u c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh x2 +y2 +z2 \u22124x+2y \u22126z \u22122 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 83 (C\u00e2u 50 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho b\u1ed1n \u0111i\u1ec3m A(1; \u02d82; 0), B(0; \u02d81; 1), C(2; 1; \u02d81) v\u00e0 D(3; 1; 4). H\u1ecfi c\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau m\u1eb7t ph\u1eb3ng c\u00e1ch \u0111\u1ec1u b\u1ed1n \u0111i\u1ec3m \u0111\u00f3? A 1 m\u1eb7t ph\u1eb3ng. B 4 m\u1eb7t ph\u1eb3ng. C 7 m\u1eb7t ph\u1eb3ng. D C\u00f3 v\u00f4 s\u1ed1 m\u1eb7t ph\u1eb3ng. \u0253 L\u1eddi gi\u1ea3i. Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (ABC) ta \u0111\u01b0\u1ee3c (ABC): x + z \u2212 1 = 0. Ki\u1ec3m tra t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m D ta suy ra 4 \u0111i\u1ec3m A; B; C; D kh\u00f4ng \u0111\u1ed3ng ph\u1eb3ng. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 522 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng c\u00e1ch \u0111\u1ec1u 4 \u0111i\u1ec3m ta c\u00f3 2 tr\u01b0\u1eddng h\u1ee3p: + Tr\u01b0\u1eddng h\u1ee3p 1 (c\u00f3 1 \u0111i\u1ec3m n\u1eb1m kh\u00e1c ph\u00eda v\u1edbi 3 \u0111i\u1ec3m c\u00f2n l\u1ea1i): c\u00f3 4 m\u1eb7t ph\u1eb3ng. + Tr\u01b0\u1eddng h\u1ee3p 2 (m\u1ed7i ph\u00eda c\u00f3 2 \u0111i\u1ec3m): c\u00f3 C32 = 3 m\u1eb7t ph\u1eb3ng. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 84 (C\u00e2u 48 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A(1; 2; 1), B(3; \u22121; 1) v\u00e0 C(\u22121; \u22121; 1). G\u1ecdi (S1) l\u00e0 m\u1eb7t c\u1ea7u c\u00f3 t\u00e2m A, b\u00e1n k\u00ednh b\u1eb1ng 2; (S2) v\u00e0 (S3) l\u00e0 hai m\u1eb7t c\u1ea7u c\u00f3 t\u00e2m l\u1ea7n l\u01b0\u1ee3t l\u00e0 B, C v\u00e0 b\u00e1n k\u00ednh \u0111\u1ec1u b\u1eb1ng 1. H\u1ecfi c\u00f3 bao nhi\u00eau m\u1eb7t ph\u1eb3ng ti\u1ebfp x\u00fac v\u1edbi c\u1ea3 ba m\u1eb7t c\u1ea7u (S1), (S2) v\u00e0 (S3) A 5. B 7. C 6. D 8. \u0253 L\u1eddi gi\u1ea3i. \u221a X\u00e9t (P ) l\u00e0 m\u1eb7t ph\u1eb3ng ti\u1ebfp x\u00fac v\u1edbi c\u1ea3 ba m\u1eb7t c\u1ea7u tr\u00ean. Ta c\u00f3 AB = AC = 13, BC = 4. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m BC; J, K l\u00e0 c\u00e1c \u0111i\u1ec3m th\u1ecfa m\u00e3n A# J\u00bb = 2 A# B\u00bb, A# K\u00bb = 2 A# C\u00bb. Ta c\u00f3 I(1; \u22121; 1), J \u00c57 \u00e3 K \u00c5 1 ; 0; \u00e3 ; 0; 1 , \u2212 1. 33 33 Ta c\u00f3 c\u00e1c tr\u01b0\u1eddng h\u1ee3p sau C\u00e1c \u0111i\u1ec3m A, B, C n\u1eb1m c\u00f9ng ph\u00eda so v\u1edbi (P ): C\u00f3 2 m\u1eb7t ph\u1eb3ng. Hai \u0111i\u1ec3m B, C n\u1eb1m c\u00f9ng ph\u00eda so v\u1edbi (P ) v\u00e0 A, B n\u1eb1m kh\u00e1c ph\u00eda so v\u1edbi (P ). Tr\u01b0\u1eddng h\u1ee3p n\u00e0y ta th\u1ea5y (P ) \u0111i qua hai \u0111i\u1ec3m J, K. Khi \u0111\u00f3 m\u1eb7t ph\u1eb3ng (P ) : by + c(z \u2212 1) = 0. Ta c\u00f3 d(B, (P )) = 1 n\u00ean \u221a| \u2212 b| = 1 \u21d2 c = 0 n\u00ean (P ) : y = 0. b2 + c2 Hai \u0111i\u1ec3m A, C n\u1eb1m c\u00f9ng ph\u00eda v\u00e0 kh\u00e1c ph\u00eda \u0111i\u1ec3m B so v\u1edbi (P ). Tr\u01b0\u1eddng h\u1ee3p n\u00e0y ta th\u1ea5y (P ) \u0111i qua I, J. l\u00e0 t\u01b0\u01a1ng t\u1ef1 nh\u01b0 tr\u00ean ta c\u00f3 2 m\u1eb7t ph\u1eb3ng. Hai \u0111i\u1ec3m A, B n\u1eb1m c\u00f9ng ph\u00eda v\u00e0 kh\u00e1c ph\u00eda \u0111i\u1ec3m C so v\u1edbi (P ). Ta c\u00f3 2 m\u1eb7t ph\u1eb3ng. V\u1eady c\u00f3 t\u1ea5t c\u1ea3 7 m\u1eb7t ph\u1eb3ng. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 85 (C\u00e2u 47 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(2; 1; 1). G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng ch\u1ee9a tr\u1ee5c Oy sao cho kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn (P ) l\u1edbn nh\u1ea5t. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (P ) l\u00e0 A x + z = 0. B x \u2212 z = 0. C 2x + z = 0. D 2x \u2212 z = 0. \u0253 L\u1eddi gi\u1ea3i. A y H K PO G\u1ecdi H v\u00e0 K l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a A tr\u00ean (P ) v\u00e0 tr\u1ee5c Oy. Ta c\u00f3 d A, (P ) = AH \u2264 AK. Do \u0111\u00f3 kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn (P ) l\u1edbn nh\u1ea5t khi H \u2261 K(0; 1; 0). #\u00bb Khi \u0111\u00f3 (P ) \u0111i qua K(0; 1; 0) v\u00e0 c\u00f3 m\u1ed9t vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 AK = (\u22122; 0; \u22121) = \u2212(2; 0; 1) n\u00ean c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 2x + z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 523 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0104 C\u00e2u 86 (C\u00e2u 44 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). \uf8f1x = 1 + 3t \uf8f4 \uf8f2 Trong kh\u00f4ng gian Oxyz cho \u0111\u01b0\u1eddng th\u1eb3ng d : y = \u22123 . G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m \uf8f4\uf8f3z = 5 + 4t A(1; \u22123; 5) v\u00e0 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = (1; 2; \u22122). \u0110\u01b0\u1eddng ph\u00e2n gi\u00e1c g\u00f3c nh\u1ecdn t\u1ea1o b\u1edfi hai \u0111\u01b0\u1eddng th\u1eb3ng d v\u00e0 \u2206 l\u00e0 \uf8f1x = \u22121 + 2t \uf8f1x = 1 + 7t \uf8f1x = 1 \u2212 t \uf8f1x = \u22121 + 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 B y = 2 \u2212 5t . C y = 3 \u2212 5t . D y = \u22123 . A y = 2 \u2212 5t . \uf8f4\uf8f3z = 6 + 11t \uf8f3\uf8f4z = \u22126 + 11t \uf8f4\uf8f3z = 5 + t \uf8f4\uf8f3z = 5 + 7t \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 \u0111i\u1ec3m A(1; \u22123; 5) thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng d n\u00ean A l\u00e0 giao \u0111i\u1ec3m c\u1ee7a d v\u00e0 \u2206. M\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 #v\u00bb = (\u22123; 0; \u22124). #\u00bb = 1 #u\u00bb = \u00c51 2 ; \u2212 2 \u00e3 #\u00bb = 1 #v\u00bb = \u00c5 3 ; 0; \u2212 4 \u00e3 Ta c\u00f3 #\u00bb \u00b7 #\u00bb > 0 n\u00ean g\u00f3c t\u1ea1o b\u1edfi hai v\u00e9c-t\u01a1 \u0110\u1eb7t u #u\u00bb| ; , v | #v\u00bb| \u2212 . u v #\u00bb #\u00bb | 33 55 3 u , v l\u00e0 g\u00f3c nh\u1ecdn t\u1ea1o b\u1edfi d v\u00e0 \u2206. Suy ra w#\u00bb = #\u00bb #\u00bb = \u00c5 4 ; 10; \u221222\u00e3 = \u2212 2 (2; \u22125; 11) l\u00e0 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c u+ v \u2212 15 15 15 15 c\u1ea7n t\u00ecm. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ea7n t\u00ecm l\u00e0 \uf8f1x = 1 + 2t \uf8f4 \uf8f2 y = \u22123 \u2212 5t \uf8f4\uf8f3z = 5 + 11t. Ch\u1ecdn t = \u22122 suy ra \u0111i\u1ec3m M (\u22121; 2; \u22126) thu\u1ed9c \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c. Khi \u0111\u00f3, \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \uf8f1x = \u22121 + 2t \uf8f4 \uf8f2 y = 2 \u2212 5t \uf8f4\uf8f3z = \u22126 + 11t. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 87 (C\u00e2u 49 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 3)2 + (y \u2212 2)2 + (z \u2212 1)2 = 1. C\u00f3 bao nhi\u00eau \u0111i\u1ec3m M thu\u1ed9c (S) sao cho ti\u1ebfp di\u1ec7n c\u1ee7a (S) t\u1ea1i M c\u1eaft tr\u1ee5c Ox, Oy l\u1ea7n l\u01b0\u1ee3t t\u1ea1i c\u00e1c \u0111i\u1ec3m A(a; 0; 0), B(0; b; 0) m\u00e0 a, b l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean d\u01b0\u01a1ng v\u00e0 A\u00f7M B = 90\u25e6. A 2. B 1. C 3. D 4. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 524 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(3; 2; 1) v\u00e0 I b\u00e1n k\u00ednh R = 1. Ta c\u00f3 IA2 = (a \u2212 3)2 + 22 + 12 = a2 \u2212 6a + 14; IB2 = 32 + (b \u2212 2)2 + 12 = b2 \u2212 4b + 14. G\u1ecdi M l\u00e0 ti\u1ebfp \u0111i\u1ec3m th\u1ecfa m\u00e3n b\u00e0i to\u00e1n IM = R = 1. M AB V\u00ec ti\u1ebfp di\u1ec7n c\u1ee7a m\u1eb7t c\u1ea7u (S) t\u1ea1i M c\u1eaft tr\u1ee5c Ox, Oy l\u1ea7n l\u01b0\u1ee3t t\u1ea1i c\u00e1c \u0111i\u1ec3m A, B n\u00ean ta c\u00f3 I\u2019M A = I\u2019M B = 90\u25e6. Suy ra M A2 = IA2 \u2212 IM 2 = a2 \u2212 6a + 13; M B2 = IB2 \u2212 IM 2 = b2 \u2212 4b + 13. Ta l\u1ea1i c\u00f3 AB2 = a2 + b2 v\u00e0 A\u00f7M B = 90\u25e6 n\u00ean AB2 = M A2 + M B2. hay a2 + b2 = a2 \u2212 6a + 13 + b2 \u2212 4b + 13 \u21d2 3a + 2b = 13. M\u1eb7t kh\u00e1c v\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean d\u01b0\u01a1ng cho n\u00ean 0 < a \u2264 4; 0 < b \u2264 5. Ta c\u00f3 b\u1ea3ng gi\u00e1 tr\u1ecb c\u1ee7a a v\u00e0 b t\u01b0\u01a1ng \u1ee9ng nh\u01b0 d\u01b0\u1edbi a1 2 3 4 b 5 3,5 2 \u22121 L\u1ea5y Lo\u1ea1i L\u1ea5y Lo\u1ea1i Th\u1eed l\u1ea1i Tr\u01b0\u1eddng h\u1ee3p 1: A(1; 0; 0) v\u00e0 B(0; 5; 0). G\u1ecdi (P ) l\u00e0 ti\u1ebfp di\u1ec7n c\u1ee7a (S) \u0111i qua A, B c\u1eaft Oz t\u1ea1i C(0; 0; c), c = 0 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh xyz (P ) : + + \u2212 1 = 0. 15c 3+ 2+ 1\u22121 (P ) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) n\u00ean 15c = 1 \u21d4 144 + 24 + 1 26 1 \u21d4 c = 60 \u2026 11 =+ . 25 5c c2 25 c2 59 1+ + 25 c2 Nh\u01b0 v\u1eady tr\u01b0\u1eddng h\u1ee3p n\u00e0y c\u00f3 1 \u0111i\u1ec3m M th\u1ecfa m\u00e3n. Tr\u01b0\u1eddng h\u1ee3p 2: A(3; 0; 0) v\u00e0 B(0; 2; 0). G\u1ecdi (P ) l\u00e0 ti\u1ebfp di\u1ec7n c\u1ee7a (S) \u0111i qua A, B c\u1eaft Oz t\u1ea1i C(0; 0; c), c = 0 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh (P ) : x + y + z \u2212 1 = 0. 32c 1 1+1+ \u22121 c \u21d2 2 1 13 1 \u21d2 72 (P ) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) n\u00ean = 1 1+ + = + c = \u2212. \u20261 1 1 c c2 36 c2 23 9 + 4 + c2 Nh\u01b0 v\u1eady tr\u01b0\u1eddng h\u1ee3p n\u00e0y c\u00f3 1 \u0111i\u1ec3m M th\u1ecfa m\u00e3n. V\u1eady c\u00f3 t\u1ea5t c\u1ea3 2 \u0111i\u1ec3m M th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 525 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0104 C\u00e2u 88 (C\u00e2u 50 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 2)2 + (y \u2212 3)2 + (z + 1)2 = 1. C\u00f3 bao nhi\u00eau \u0111i\u1ec3m M thu\u1ed9c (S) sao cho ti\u1ebfp di\u1ec7n c\u1ee7a (S) t\u1ea1i M c\u1eaft tr\u1ee5c Ox, Oy l\u1ea7n l\u01b0\u1ee3t t\u1ea1i c\u00e1c \u0111i\u1ec3m A(a; 0; 0), B(0; b; 0) m\u00e0 a, b l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean d\u01b0\u01a1ng v\u00e0 A\u00f7M B = 90\u25e6? A 1. B 2. C 4. D 3. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(2; 3; \u22121) v\u00e0 b\u00e1n k\u00ednh R = 1. y M Ta c\u00f3 B I IA2 = (a \u2212 2)2 + (\u22123)2 + 12 = a2 \u2212 4a + 14. IB2 = (\u22122)2 + (b \u2212 3)2 + 12 = b2 \u2212 6b + 14. G\u1ecdi M l\u00e0 \u0111i\u1ec3m th\u1ecfa m\u00e3n b\u00e0i to\u00e1n th\u00ec IM = R = 1. O x V\u00ec ti\u1ebfp di\u1ec7n c\u1ee7a (S) t\u1ea1i M c\u1eaft tr\u1ee5c Ox, Oy l\u1ea7n l\u01b0\u1ee3t t\u1ea1i c\u00e1c \u0111i\u1ec3m A, B n\u00ean IM \u22a5 (M AB), suy ra IM \u22a5 M A v\u00e0 IM \u22a5 M B. A IM A vu\u00f4ng t\u1ea1i M n\u00ean M A2 = IA2 \u2212 IM 2 = a2 \u2212 4a + 13. IM B vu\u00f4ng t\u1ea1i M n\u00ean M B2 = IB2 \u2212 IM 2 = b2 \u2212 6b + 13. AM B vu\u00f4ng t\u1ea1i M n\u00ean AB2 = M A2 + M B2 \u21d4 a2 + b2 = a2 \u2212 4a + 13 + b2 \u2212 6b + 13 \u21d4 2a + 3b = 13. M\u1eb7t kh\u00e1c a, b l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean d\u01b0\u01a1ng n\u00ean ta c\u00f3 c\u00e1c tr\u01b0\u1eddng h\u1ee3p sau Tr\u01b0\u1eddng h\u1ee3p 1. a = 5, b = 1. Th\u1eed l\u1ea1i A(5; 0; 0), B(0; 1; 0). N\u1ebfu m\u1eb7t ph\u1eb3ng (P ) \u0111i qua A, B v\u00e0 song song v\u1edbi Oz th\u00ec (P ) : x + 5y \u2212 5 = 0. Khi \u0111\u00f3 m\u1eb7t ph\u1eb3ng P kh\u00f4ng ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) c\u00f3 d(I, (P )) = \u221a12 > 1. 26 G\u1ecdi (P ) l\u00e0 ti\u1ebfp di\u1ec7n c\u1ee7a (S) \u0111i qua A, B c\u1eaft Oz t\u1ea1i C(0; 0; c), c = 0, c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh (P ) : x + y + z \u2212 1 = 0. 5c (P ) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) n\u00ean 2 +3\u2212 1 \u22121 = 1 \u21d2 144 \u2212 24 + 1 1 1 60 5 c . = +1+ \u21d2 c = \u20261 1 25 5c c2 25 c2 59 +1+ 25 c2 Ch\u00fa \u00fd r\u1eb1ng qua A, B c\u00f2n c\u00f3 m\u1eb7t ph\u1eb3ng (Oxy) c\u0169ng ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) nh\u01b0ng ti\u1ebfp di\u1ec7n n\u00e0y kh\u00f4ng th\u1ecfa m\u00e3n b\u00e0i to\u00e1n. Nh\u01b0 v\u1eady, tr\u01b0\u1eddng h\u1ee3p n\u00e0y c\u00f3 1 \u0111i\u1ec3m M th\u1ecfa m\u00e3n. Tr\u01b0\u1eddng h\u1ee3p 2. a = 2, b = 3. Th\u1eed l\u1ea1i A(2; 0; 0), B(0; 3; 0). N\u1ebfu m\u1eb7t ph\u1eb3ng (P ) \u0111i qua A, B v\u00e0 song song v\u1edbi Oz th\u00ec (P ) : 3x + 2y \u2212 6 = 0. Khi \u0111\u00f3 m\u1eb7t ph\u1eb3ng P kh\u00f4ng ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) c\u00f3 d(I, (P )) = \u221a6 > 1. 13 G\u1ecdi (P ) l\u00e0 ti\u1ebfp di\u1ec7n c\u1ee7a (S) \u0111i qua A, B c\u1eaft Oz t\u1ea1i C(0; 0; c), c = 0, c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh (P ) : x + y + z \u2212 1 = 0. 23c Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 526 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN (P ) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) n\u00ean 1+1\u2212 1 \u22121 =1\u21d21\u2212 2 + 1 13 1 72 c = +1+ . \u21d2c= \u20261 1 1 c c2 36 c2 23 4 + 9 + c2 Ch\u00fa \u00fd r\u1eb1ng qua A, B c\u00f2n c\u00f3 m\u1eb7t ph\u1eb3ng (Oxy) c\u0169ng ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) nh\u01b0ng ti\u1ebfp di\u1ec7n n\u00e0y kh\u00f4ng th\u1ecfa m\u00e3n b\u00e0i to\u00e1n. Nh\u01b0 v\u1eady, tr\u01b0\u1eddng h\u1ee3p n\u00e0y c\u00f3 1 \u0111i\u1ec3m M th\u1ecfa m\u00e3n. V\u1eady c\u00f3 2 \u0111i\u1ec3m M th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 89 (C\u00e2u 50 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(1; \u22123; 2) v\u00e0 B(\u22122; 1; \u22123). X\u00e9t hai \u0111i\u1ec3m M v\u00e0 N thay \u0111\u1ed5i thu\u221a\u1ed9c m\u1eb7t ph\u1eb3ng (Oxy) sa\u221ao cho M N = 1. Gi\u00e1 tr\u1ecb \u221al\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN\u221a| b\u1eb1ng A 17. B 41. C 37. D 61. \u0253 L\u1eddi gi\u1ea3i. Ta th\u1ea5y A v\u00e0 B n\u1eb1m kh\u00e1c ph\u00eda so v\u1edbi m\u1eb7t ph\u1eb3ng A B (Oxy). B1 G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi m\u1eb7t A ph\u1eb3ng (Oxy) \u21d2 (P ) : z = 2. (P ) G\u1ecdi B l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi B qua m\u1eb7t ph\u1eb3ng (Oxy) \u21d2 B (\u22122; 1; 3). B1 l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a B tr\u00ean M N m\u1eb7t ph\u1eb3ng (P ) \u21d2 B1(\u22122; 1; 2). D\u1ef1ng h\u00ecnh b\u00ecnh h\u00e0nh (Oxy) AM N A . Khi \u0111\u00f3 AA = 1 v\u00e0 AA \u2225 (Oxy). Suy ra A thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n (C) c\u00f3 t\u00e2m A v\u00e0 b\u00e1n k\u00ednh R = 1, (C) n\u1eb1m trong m\u1eb7t ph\u1eb3ng (P ). B Ta c\u00f3 |AM \u2212 BN | = |A N \u2212 BN | = |A N \u2212 N B | \u2264 AB. Ta th\u1ea5y AB1 = 5 > R \u21d2 B1 n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n (C). Do A \u2208 (P ), B \u2208\/ (P ) m\u00e0 (P ) \u2225 (Oxy) suy ra A B lu\u00f4n c\u1eaft m\u1eb7t ph\u1eb3ng (Oxy). Ta c\u00f3 A B = B1B 2 + A B12 = 1 + A B12, do \u0111\u00f3 A B l\u1edbn nh\u1ea5t khi v\u00e0 ch\u1ec9 khi A B1 l\u1edbn nh\u1ea5t. Ta l\u1ea1i c\u00f3 A B1 \u2264 AB1 + R = 6. \u0110\u1eb3ng th\u1ee9c x\u1ea3y ra khi A l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AB1 v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (C) (A \u1edf gi\u1eefa A v\u00e0 B1) v\u00e0 N l\u00e0 giao \u0111i\u1ec3m c\u1ee7a A B\u221a v\u1edbi m\u1eb7t p\u221ah\u1eb3ng (Oxy). V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN | b\u1eb1ng 1 + 62 = 37. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 90 (C\u00e2u 49 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(1; \u22123; 2) v\u00e0 B(\u22122; 1; \u22124). X\u00e9t hai \u0111i\u1ec3m M v\u00e0 N thay Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 527 S\u0110T: 0905.193.688","2. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng \u0111\u1ed5i thu\u1ed9\u221ac m\u1eb7t ph\u1eb3ng (Oxy) sao\u221acho M N = 4. Gi\u00e1 tr\u1ecb \u221al\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN\u221a| b\u1eb1ng A 5 2. B 3 13. C 61. D 85. \u0253 L\u1eddi gi\u1ea3i. A B A B1 N M B D\u1ec5 th\u1ea5y hai \u0111i\u1ec3m A, B n\u1eb1m kh\u00e1c ph\u00eda so v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy). G\u1ecdi B l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi B qua m\u1eb7t ph\u1eb3ng (Oxy), khi \u0111\u00f3 B (\u22122; 1; 4). G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy), ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) l\u00e0 (P ) : z = 2. G\u1ecdi B1 l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4#ng\u00bbg\u00f3c #c\u1ee7a\u00bb B tr\u00ean m\u1eb7t ph\u1eb3ng (P ), khi \u0111\u00f3 B1(\u22122; 1; 2). G\u1ecdi A l\u00e0 \u0111i\u1ec3m sao cho AA = M N , khi \u0111\u00f3 AA = M N = 4 v\u00e0 AM = A N . \u0110i\u1ec3m A thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n (C) n\u1eb1m trong m\u1eb7t ph\u1eb3ng (P ), c\u00f3 t\u00e2m A b\u00e1n k\u00ednh R = 4. Ta c\u00f3 |AM \u2212 BN | = |A N \u2212 B N | \u2264 A B . D\u1ea5u \u201c=\u201d x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi N l\u00e0 giao \u0111i\u1ec3m c\u1ee7a A B v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy) v\u00e0 N n\u1eb1m ngo\u00e0i \u0111o\u1ea1n th\u1eb3ng A B . Ta th\u1ea5y A , B n\u1eb1m c\u00f9ng m\u1ed9t ph\u00eda v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy), A \u2208 (P ), B \u2208\/ (P ), (P ) \u2225 (Oxy) n\u00ean A B lu\u00f4n c\u1eaft (Oxy) t\u1ea1i \u0111i\u1ec3m n\u1eb1m ngo\u00e0i \u0111o\u1ea1n A B . Ta c\u00f3 B B1 = 2; AB1 = 5 n\u00ean suy ra A B1 \u2264 AB1 + A A = 5 + 4 = 9. Do \u0111\u00f3 \u00bb \u221a\u221a A B = A B12 + B B12 \u2264 92 + 22 = 85. D\u1ea5u \u201c=\u201d x\u1ea3y ra khi A l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AB1 v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (C) (A \u1edf gi\u1eefa A v\u00e0 B1 v\u00e0 N l\u00e0 giao \u0111i\u1ec3m c\u1ee7a A B v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy)). \u221a V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN | b\u1eb1ng 85. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 91 (C\u00e2u 46 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; 2). G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng ch\u1ee9a tr\u1ee5c Ox sao cho kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn (P ) l\u1edbn nh\u1ea5t. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (P ) l\u00e0: A 2y \u2212 z = 0. B 2y + z = 0. C y \u2212 z = 0. D y + z = 0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(1; 2; 2) l\u00ean tr\u1ee5c Ox l\u00e0 M (1; 0; 0). #\u00bb Kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn (P ) l\u1edbn nh\u1ea5t n\u00ean m\u1eb7t ph\u1eb3ng (P ) c\u00f3 vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 M# A\u00bb= (0; 2; 2). Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) \u0111i qua \u0111i\u1ec3m M (1; 0; 0) v\u00e0 c\u00f3 vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 M A = (0; 2; 2) n\u00ean 0.(x \u2212 1) + 2(y \u2212 0) + 2(z \u2212 0) = 0 \u21d4 y + z = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 528 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 92 (C\u00e2u 47 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111i\u1ec3m A(4; 6; 2), B(2; \u22122; 0) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x + y + z = 0. X\u00e9t \u0111\u01b0\u1eddng th\u1eb3ng d thay \u0111\u1ed5i thu\u1ed9c (P ) v\u00e0 \u0111i qua B, g\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A tr\u00ean d. Bi\u1ebft r\u1eb1ng khi d thay \u0111\u1ed5i th\u00ec H thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n c\u1ed1 \u0111\u1ecbnh. T\u00ednh b\u00e1n k\u00ednh R c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n \u221a\u0111\u00f3. B R = 2. C R = 1. \u221a A R = 6. D R = 3. \u0253 L\u1eddi gi\u1ea3i. \u221a - M\u1eb7t c\u1ea7u \u0111\u01b0\u1eddng k\u00ednh AB c\u00f3 t\u00e2m I(3; 2; 1) v\u00e0 b\u00e1n k\u00ednh R = 18. - H lu\u00f4n thu\u1ed9c m\u1eb7t ph\u1eb3ng (P ) v\u00e0 m\u1eb7\u221at c\u1ea7u \u0111\u01b0\u1eddng k\u00ednh AB.\u221a - Kho\u1ea3ng c\u00e1ch t\u1eeb I \u0111\u1ebfn (P ) l\u00e0 d = 2 3. T\u1eeb \u0111\u00f3 suy ra R = 6. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 93 (C\u00e2u 49 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111i\u1ec3m A(3; \u22122; 6), B(0; 1; 0) v\u00e0 m\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + (y \u2212 2)2 + (z \u2212 3)2 = 25. M\u1eb7t ph\u1eb3ng (P ) : ax + by + cz \u2212 2 = 0 \u0111i qua A, B v\u00e0 c\u1eaft (S) theo giao tuy\u1ebfn l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 b\u00e1n k\u00ednh nh\u1ecf nh\u1ea5t. T\u00ednh T = a + b + c. A T = 3. B T = 5. C T = 2. D T = 4. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec3m A n\u1eb1m b\u00ean ngo\u00e0i m\u1eb7t c\u1ea7u (S) v\u00e0 \u0111i\u1ec3m B n\u1eb1m b\u00ean trong m\u1eb7t c\u1ea7u (S). Do \u0111\u00f3 \u0111\u1ec3 (P )\u0111\u1ebfcn\u1eaft(P(S))pthh\u1ea3eiol\u1edbgnianoht\u1ea5uty, \u1ebftn\u1ee9cl\u00e0l\u00e0\u0111I\u01b0#M\u1edd\u00bbngphtr\u1ea3\u00f2inl\u00e0c\u00f3v\u00e9bct\u00e1\u01a1n k\u00ednh nh\u1ecf nh\u1ea5t th\u00ec kho\u1ea3ng c\u00e1ch t\u1eeb t\u00e2m I c\u1ee7a m\u1eb7t c\u1ea7u ph\u00e1p tuy\u1ebfn c\u1ee7a (P ), v\u1edbi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng CD, v\u1edbi C v\u00e0 D l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng AB v\u1edbi m\u1eb7t c\u1ea7u (S). VI#Mi\u1ebf\u00bbt=ph(\u01b00;\u01a1\u2212ng2;t\u2212r\u00ecn1)h. \u0111\u01b0\u1eddng th\u1eb3ng AB, t\u00ecm giao \u0111i\u1ec3m v\u1edbi m\u1eb7t c\u1ea7u, sau \u0111\u00f3 ta t\u00ecm \u0111\u01b0\u1ee3c M (1; 0; 2) suy ra Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) qua B(0; 1; 0) l\u00e0 0(x \u2212 0) \u2212 2(y \u2212 1) \u2212 1(z \u2212 0) = 0 \u21d4 2y + z \u2212 2 = 0 Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 529 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian B\u00c0I 3. PH\u01af\u01a0NG TR\u00ccNH \u0110\u01af\u1edcNG TH\u1eb2NG TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 1 (C\u00e2u 44 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). \uf8f1x = 1 \uf8f4 \uf8f2 Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : y = 2 + 3t (t \u2208 R). Vect\u01a1 n\u00e0o d\u01b0\u1edbi \uf8f4\uf8f3z = 5 \u2212 t \u0111\u00e2y l\u00e0 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d ? A u#\u00bb1 = (0; 3; \u22121). B u#\u00bb2 = (1; 3; \u22121). C u#\u00bb3 = (1; \u22123; \u22121). D u#\u00bb4 = (1; 2; 5). \u0253 L\u1eddi gi\u1ea3i. V\u00e9c t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 u#\u00bb1 = (0; 3; \u22121). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 2 (C\u00e2u 8 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). \uf8f1x = 2 \u2212 t \uf8f4 \uf8f2 Trong kh\u00f4ng gian Oxyz, \u0111\u01b0\u1eddng th\u1eb3ng d : y = 1 + 2t c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 \uf8f4\uf8f3z = 3 + t A #u\u00bb3 = (2; 1; 3). B #u\u00bb4 = (\u22121; 2; 1). C #u\u00bb2 = (2; 1; 1). D #u\u00bb1 = (\u22121; 2; 3). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb4 = (\u22121; 2; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 3 (C\u00e2u 14 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). x+3 y\u22121 z\u22125 Trong kh\u00f4ng gian Oxyz, \u0111\u01b0\u1eddng th\u1eb3ng d : 1 = \u22121 = 2 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 A #u\u00bb1 = (3; \u22121; 5). B #u\u00bb4 = (1; \u22121; 2). C #u\u00bb2 = (\u22123; 1; 5). D #u\u00bb3 = (1; \u22121; \u22122). \u0253 L\u1eddi gi\u1ea3i. M\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 #u\u00bb = (1; \u22121; 2). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 4 (C\u00e2u 7 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). x\u22122 y\u22121 z+3 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : \u22121 = = . V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t 2 1 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d? C #u\u00bb3 = (\u22121; 2; 1). A #u\u00bb2 = (2; 1; 1). B #u\u00bb4 = (1; 2; \u22123). D #u\u00bb1 = (2; 1; \u22123). \u0253 L\u1eddi gi\u1ea3i. M\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d l\u00e0 #u\u00bb3 = (\u22121; 2; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 5 (C\u00e2u 9 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). x\u22121 y\u22123 z+2 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : = = . V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 2 \u22125 3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d C #u\u00bb = (1; 3; 2). D #u\u00bb = (1; 3; \u22122). A #u\u00bb = (2; 5; 3). B #u\u00bb = (2; \u22125; 3). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 530 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0253 L\u1eddi gi\u1ea3i. D\u1ef1a v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng suy ra m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d l\u00e0 #u\u00bb = (2; \u22125; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 6 (C\u00e2u 13 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). x+2 y\u22121 z\u22123 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : = \u22123 = . Vec-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t 1 2 vec-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d? B u#\u00bb3 = (\u22122; 1; 3). C u#\u00bb1 = (\u22122; 1; 2). D u#\u00bb4 = (1; 3; 2). A u#\u00bb2 = (1; \u22123; 2). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d: x+2 = y\u22121 = z \u22123 c\u00f3 m\u1ed9t vec-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 u#\u00bb2 = (1; \u22123; 2). 1 \u22123 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 7 (C\u00e2u 11 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). x\u22123 y+1 z\u22125 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : = \u22122 = . V\u00e9c-t\u01a1 n\u00e0o sau \u0111\u00e2y l\u00e0 m\u1ed9t 1 3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d? A u#\u00bb1 = (3; \u22121; 5). B u#\u00bb3 = (2; 6; \u22124). C u#\u00bb4 = (\u22122; \u22124; 6). D u#\u00bb2 = (1; \u22122; 3). \u0253 L\u1eddi gi\u1ea3i. Ta th\u1ea5y \u0111\u01b0\u1eddng th\u1eb3ng d c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 t\u1ecda \u0111\u1ed9 u#\u00bb2 = (1; \u22122; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 8 (C\u00e2u 19 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). x\u22123 y\u22124 z+1 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : = \u22125 = . V\u00e9c-t\u01a1 n\u00e0o sau \u0111\u00e2y l\u00e0 m\u1ed9t 2 3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d? C #u\u00bb3 = (2; 5; 3). A #u\u00bb2 = (3; 4; \u22121). B #u\u00bb1 = (2; \u22125; 3). D #u\u00bb4 = (3; 4; 1). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh d\u1ea1ng x \u2212 x0 = y \u2212 y0 = z \u2212 z0 th\u00ec c\u00f3 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (a; b; c). abc x\u22123 y\u22124 z +1 N\u00ean \u0111\u01b0\u1eddng th\u1eb3ng d : 2 = \u22125 = 3 c\u00f3 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb1 = (2; \u22125; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 9 (C\u00e2u 19 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). x\u22122 y+5 z\u22122 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : 3 = 4 = \u22121 . V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d? A #u\u00bb2 = (3; 4; \u22121). B #u\u00bb1 = (2; \u22125; 2). C #u\u00bb3 = (2; 5; \u22122). D #u\u00bb4 = (3; 4; 1). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb2 = (3; 4; \u22121). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 10 (C\u00e2u 3 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). x\u22124 y+2 z\u22123 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : 3 = \u22121 = \u22122 . V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 531 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian A #u\u00bb2 = (4; \u22122; 3). B #u\u00bb4 = (4; 2; \u22123). C #u\u00bb3 = (3; \u22121; \u22122). D #u\u00bb1 = (3; 1; 2). \u0253 L\u1eddi gi\u1ea3i. V\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 #u\u00bb3 = (3; \u22121; \u22122). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 11 (C\u00e2u 23 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). \uf8f1x = 2 + t \uf8f4 \uf8f2 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : y = 1 \u2212 2t. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 \uf8f3\uf8f4z = \u22121 + 3t ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d? B #u\u00bb2 = (1; 2; 3). C #u\u00bb3 = (1; \u22122; 3). D #u\u00bb4 = (2; 1; 1). A #u\u00bb1 = (2; 1; \u22121). \u0253 L\u1eddi gi\u1ea3i. T\u1eeb ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d ta th\u1ea5y v\u00e9c-t\u01a1 #u\u00bb3 = (1; \u22122; 3) l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 12 (C\u00e2u 27 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). \uf8f1x = 2 + t \uf8f4 \uf8f2 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng (d) : y = 1 \u2212 2t V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 \uf8f3\uf8f4z = \u22121 + 3t. ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d? B u#\u00bb1 = (2; 1; \u22121). C u#\u00bb3 = (1; \u22122; 3). D u#\u00bb3 = (1; 2; 3). A u#\u00bb4 = (2; 1; 1). \u0253 L\u1eddi gi\u1ea3i. M\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 u#\u00bb3 = (1; \u22122; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 13 (C\u00e2u 37 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (1; 2; \u22122) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x + y \u2212 3x + 1 = 0. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua M vu\u00f4ng g\u00f3c v\u1edbi (P ) l\u00e0 \uf8f1x = \u22121 + 2t \uf8f1x = 1 + 2t \uf8f1x = 1 \u2212 2t \uf8f1x = 2 + t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22122 + t B y=2+t C y=2+t D y = 1 + 2t \uf8f4\uf8f3y = 2 \u2212 3t. \uf8f3\uf8f4z = \u22122 \u2212 3t. \uf8f4\uf8f3z = \u22122 \u2212 3t. \uf8f3\uf8f4z = \u22123 \u2212 2t. \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm nh\u1eadn v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn #n\u00bb(2; 1; \u22123) c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng, \uf8f1x = 1 + 2t \uf8f4 \uf8f2 do \u0111\u00f3 n\u00f3 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh y = 2 + t . \uf8f4\uf8f3z = \u22122 \u2212 3t Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 14 (C\u00e2u 4 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m M (3; \u22121; 4) v\u00e0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (\u22122; 4; 5). Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a d l\u00e0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 532 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \uf8f1x = \u22122 + 3t \uf8f1x = 3 + 2t \uf8f1x = 3 \u2212 2t \uf8f1x = 3 \u2212 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y=4\u2212t . B y = \u22121 + 4t . C y = 1 + 4t . D y = \u22121 + 4t . \uf8f3\uf8f4z = 5 + 4t \uf8f4\uf8f3z = 4 + 5t \uf8f3\uf8f4z = 4 + 5t \uf8f4\uf8f3z = 4 + 5t \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m M (3; \u22121; 4) v\u00e0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (\u22122; 4; 5) n\u00ean d c\u00f3 ph\u01b0\u01a1ng \uf8f1x = 3 \u2212 2t \uf8f4 \uf8f2 tr\u00ecnh l\u00e0 y = \u22121 + 4t \uf8f4\uf8f3z = 4 + 5t. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 15 (C\u00e2u 7 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m M (2; 2; 1) v\u00e0 c\u00f3 m\u1ed9t vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (5; 2; \u22123). Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a d l\u00e0 \uf8f1x = 2 + 5t \uf8f1x = 2 + 5t \uf8f4 \uf8f4 \uf8f2 \uf8f2 A y = 2 + 2t (t \u2208 R). B y = 2 + 2t (t \u2208 R). \uf8f3\uf8f4z = \u22121 \u2212 3t \uf8f3\uf8f4z = 1 + 3t \uf8f1x = 2 + 5t \uf8f1x = 5 + 2t \uf8f4 \uf8f4 \uf8f2 \uf8f2 C y = 2 + 2t (t \u2208 R). D y = 2 + 2t (t \u2208 R). \uf8f3\uf8f4z = 1 \u2212 3t \uf8f3\uf8f4z = \u22123 + t \u0253 L\u1eddi gi\u1ea3i. \uf8f1x = 2 + 5t \uf8f4 Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a d \u0111i qua \u0111i\u1ec3m M (2; 2; 1) v\u00e0 c\u00f3 m\u1ed9t vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (5; 2; \u22123) l\u00e0 \uf8f2 = 2 + 2t (t \u2208 y \uf8f4\uf8f3z = 1 \u2212 3t R). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 16 (C\u00e2u 5 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m M (\u22123; 1; 2) v\u00e0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (2; 4; \u22121). Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a d l\u00e0 \uf8f1x = 3 + 2t \uf8f1x = \u22123 + 2t \uf8f1x = \u22123 + 2t \uf8f1x = 2 \u2212 3t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = 1 + 4t B y = 1 + 4t C y = 1 + 4t D y=4+t \uf8f4\uf8f3z = 2 \u2212 t. \uf8f4\uf8f3z = 2 + t. \uf8f3\uf8f4z = 2 \u2212 t. \uf8f4\uf8f3z = \u22121 + 2t. \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m M (\u22123; 1; 2) v\u00e0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (2; 4; \u22121) n\u00ean c\u00f3 ph\u01b0\u01a1ng \uf8f1x = \u22123 + 2t \uf8f4 \uf8f2 tr\u00ecnh y = 1 + 4t , t \u2208 R. \uf8f4\uf8f3z = 2 \u2212 t Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 17 (C\u00e2u 20 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m M (1; 5; \u22122) v\u00e0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (3; \u22126; 1). Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a d l\u00e0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 533 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian \uf8f1x = 3 + t \uf8f1x = 1 + 3t \uf8f1x = 1 + 3t \uf8f1x = 1 + 3t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22126 + 5t . B y = 5 \u2212 6t . C y = 5 + 6t . D y = 5 \u2212 6t . \uf8f3\uf8f4z = 1 \u2212 2t \uf8f3\uf8f4z = 2 + t \uf8f4\uf8f3z = \u22122 + t \uf8f4\uf8f3z = \u22122 + t \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d \u0111i qua M (1; 5; \u22122) v\u00e0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (3; \u22126; 1) th\u00ec ph\u01b0\u01a1ng tr\u00ecnh tham \uf8f1x = 1 + 3t \uf8f4 \uf8f2 s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 y = 5 \u2212 6t \uf8f4\uf8f3z = \u22122 + t. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 18 (C\u00e2u 21 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M (\u22122; 1; 3) v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (1; \u22123; 5) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh x\u22122 y+1 z\u22123 x\u22121 y+3 z\u22125 A \u22122 = = . B = \u22123 = . 1 3 1 5 x+2 y\u22121 z\u22123 x+2 y\u22121 z\u22123 C = =. D = \u22123 = . 135 1 5 \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M (\u22122; 1; 3) v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (1; \u22123; 5) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 x+2 y\u22121 z\u22123 = =. 1 \u22123 5 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 19 (C\u00e2u 36 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m M (1; 2; 1) v\u00e0 N (3; 1; \u22122). \u0110\u01b0\u1eddng th\u1eb3ng M N c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22121 y\u22122 z\u22121 x+1 y+2 z+1 2 = \u22121 = \u22123 . A 4 = 3 = \u22121 . B C x\u22121 y\u22122 z\u22121 D x+1 y+2 z+1 4 = 3 = \u22121 . 2 = \u22121 = \u22123 . \u0110\u01b0\u1eddng th\u1eb3ng MN \u0111i qua \u0111i\u1ec3m M (1; 2; 1), \u0253 L\u1eddi gi\u1ea3i.# \u00bb = (2; \u22121; \u22123) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 nh\u1eadn v\u00e9c-t\u01a1 M N x\u22121 y\u22122 z\u22121 ph\u01b0\u01a1ng tr\u00ecnh ch\u00ednh t\u1eafc l\u00e0 2 = \u22121 = \u22123 . Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 20 (C\u00e2u 24 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M (\u22122; 1; 3) v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (2; \u22123; 4) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22122 y+1 z+3 x+2 y\u22121 z\u22123 A = \u22123 = . B = \u22123 = . 2 4 2 4 x\u22122 y+3 z\u22124 x+2 y\u22121 z\u22123 C \u22122 = = . D = =. 1 3 234 \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M (\u22122; 1; 3) v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (2; \u22123; 4) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng x+2 y\u22121 z\u22123 tr\u00ecnh l\u00e0 = \u22123 = . 2 4 Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 534 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 21 (C\u00e2u 17 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M (\u22122; 1; 3) v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (2; 3; \u22125) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22122 y+1 z+3 x+2 y\u22121 z\u22123 A 2 = 3 = \u22125 . B 2 = 3 = \u22125 . C x\u22122 y\u22123 z+5 D x+2 y\u22121 z\u22123 \u22122 = 1 = . = =. 235 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M (\u22122; 1; 3) v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (2; 3; \u22125) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng x+2 y\u22121 z\u22123 tr\u00ecnh l\u00e0 2 = 3 = \u22125 . Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 22 (C\u00e2u 19 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M (\u22122; 1; 3) v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (1; 3; \u22125) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x+2 y\u22121 z\u22123 x+2 y\u22121 z\u22123 A = =. B 1 = 3 = \u22125 . 135 x\u22121 y\u22123 z+5 x\u22122 y+1 z+3 C \u22122 = 1 = . D 1 = 3 = \u22125 . 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M (\u22122; 1; 3) v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (1; 3; \u22125) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x+2 y\u22121 z\u22123 1 = 3 = \u22125 . Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 23 (C\u00e2u 15 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). x+2 y\u22121 z+2 Trong kh\u00f4ng gian Oxyz, \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng d : = = ? 112 A P (1; 1; 2). B N (2; \u22121; 2). C Q(\u22122; 1; \u22122). D M (\u22122; \u22122; 1). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d : x + 2 = y \u2212 1 = z + 2 \u0111i qua \u0111i\u1ec3m Q(\u22122; 1; \u22122). 112 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 24 (C\u00e2u 10 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). \uf8f1x = 1 \u2212 t \uf8f4 \uf8f2 Trong kh\u00f4ng gian Oxyz, \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng d : y = 5 + t ? \uf8f3\uf8f4z = 2 + 3t A P (1; 2; 5). B N (1; 5; 2). C Q (\u22121; 1; 3). D M (1; 1; 3). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m N (1; 5; 2). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 25 (C\u00e2u 49 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). x\u22121 y\u22122 z\u22123 Trong kh\u00f4ng gian Oxyz, \u0111\u01b0\u1eddng th\u1eb3ng d : 2 = \u22121 = 2 \u0111i qua \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 535 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian A Q(2; \u22121; 2). B M (\u22121; \u22122; \u22123). C P (1; 2; 3). D N (\u22122; 1; \u22122). \u0253 L\u1eddi gi\u1ea3i. Thay l\u1ea7n l\u01b0\u1ee3t t\u1ecda \u0111\u1ed9 c\u00e1c \u0111i\u1ec3m \u0111\u00e3 cho v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d, ta c\u00f3 V\u1edbi M (\u22121; \u22122; \u22123) th\u00ec \u22121 \u2212 1 = \u22122 \u2212 2 suy ra d kh\u00f4ng \u0111i qua \u0111i\u1ec3m M. 2 \u22121 , V\u1edbi N (\u22122; 1; \u22122) th\u00ec \u22122 \u2212 1 = 1\u22122 suy ra d kh\u00f4ng \u0111i qua \u0111i\u1ec3m N. 2 \u22121 , 1\u22121 2\u22122 3\u22123 V\u1edbi P (1; 2; 3) th\u00ec == = 0, suy ra d \u0111i qua \u0111i\u1ec3m P . 2 \u22121 2 V\u1edbi Q(2; \u22121; 2) th\u00ec 2\u22121 = \u22121 \u2212 2 suy ra d kh\u00f4ng \u0111i qua \u0111i\u1ec3m Q. 2 \u22121 , Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 26 (C\u00e2u 6 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (3; \u22121; 1) tr\u00ean tr\u1ee5c Oz c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (3; 0; 0). B (3; \u22121; 0). C (0; 0; 1). D (0; \u22121; 0). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (3; \u22121; 1) tr\u00ean tr\u1ee5c Oz c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (0; 0; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 27 (C\u00e2u 25 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). x\u22121 y\u22122 z+1 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : 2 = 3 = \u22121 . \u0110i\u1ec3m n\u00e0o thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng d? A P (1; 2; \u22121). B M (\u22121; \u22122; 1). C N (2; 3; \u22121). D Q(\u22122; \u22123; 1). \u0253 L\u1eddi gi\u1ea3i. Th\u1ebf t\u1ecda \u0111\u1ed9 P (1; 2; \u22121) v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d ta c\u00f3 1\u22121 = 2\u22122 = \u22121 + 1 Do \u0111\u00f3 2 3 \u22121 . P (1; 2; \u22121) thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng d. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 28 (C\u00e2u 12 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). x\u22122 y\u22121 z+3 Trong kh\u00f4ng gian Oxyz cho \u0111\u01b0\u1eddng th\u1eb3ng d : = \u22122 = . \u0110i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c 4 1 d? A Q(4; \u22122; 1). B N (4; 2; 1). C P (2; 1; \u22123). D M (2; 1; 3). \u0253 L\u1eddi gi\u1ea3i. T\u1eeb ph\u01b0\u01a1ng tr\u00ecnh d: x\u22122 = y\u22121 = z +3 ta th\u1ea5y P (2; 1; \u22123) l\u00e0 m\u1ed9t \u0111i\u1ec3m thu\u1ed9c d. 4 \u22122 1 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 29 (C\u00e2u 5 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). x\u22124 y\u22122 z+1 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : = \u22125 = . \u0110i\u1ec3m n\u00e0o sau \u0111\u00e2y thu\u1ed9c 2 1 d? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 536 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN A N (4; 2; \u22121). B Q(2; 5; 1). C M (4; 2; 1). D P (2; \u22125; 1). \u0253 L\u1eddi gi\u1ea3i. L\u1ea7n l\u01b0\u1ee3t thay to\u1ea1 \u0111\u1ed9 c\u00e1c \u0111i\u1ec3m \u1edf c\u00e1c ph\u01b0\u01a1ng \u00e1n l\u1ef1a ch\u1ecdn v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng, ta th\u1ea5y 4 \u2212 4 = 2 \u2212 2 = \u22121 + 1 \u21d2 N \u2208 d. 25 1 2 \u2212 4 = 5\u22122 \u21d2 Q \u2208\/ d. 2 \u22125 4 \u2212 4 = 2\u22122 = 1 +1 \u21d2 M \u2208\/ d. 2 \u22125 1 2\u22124 = \u22125 \u2212 2 \u21d2 P \u2208\/ d. 2 \u22125 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 30 (C\u00e2u 21 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). x\u22123 y+1 z+2 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : 2 = 4 = \u22121 . \u0110i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c d? A N (3; \u22121; \u22122). B Q(2; 4; 1). C P (2; 4; \u22121). D M (3; 1; 2). \u0253 L\u1eddi gi\u1ea3i. Ta th\u1ea5y t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m N (3; \u22121; \u22122) th\u1ecfa m\u00e3n ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d. V\u1eady \u0111i\u1ec3m N thu\u1ed9c d. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 31 (C\u00e2u 11 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). x\u22123 y\u22121 z+5 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : 2 = 2 = \u22121 . \u0110i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c d? A M (3; 1; 5). B N (3; 1; \u22125). C P (2; 2; \u22121). D Q(2; 2; 1). \u0253 L\u1eddi gi\u1ea3i. Thay t\u1ecda \u0111\u1ed9 t\u1eebng \u0111i\u1ec3m v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d, ta th\u1ea5y N (3; 1; \u22125) \u2208 d. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 32 (C\u00e2u 20 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). x\u22122 y\u22121 z + 1\u00b7 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d: 1 = \u22122 = 3 \u0110i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c d? A Q(2; 1; 1). B M (1; 2; 3). C P (2; 1; \u22121). D N (1; \u22122; 3). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m P (2; 1; \u22121). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 33 (C\u00e2u 30 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; \u22122; 3) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d: x\u22121 = y+2 = z\u22123 M\u1eb7t 3 2 \u22121 . ph\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 3x + 2y \u2212 z + 1 = 0. B 2x \u2212 2y + 3z \u2212 17 = 0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 537 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian C 3x + 2y \u2212 z \u2212 1 = 0. D 2x \u2212 2y + 3z + 17 = 0. \u0253 L\u1eddi gi\u1ea3i. x \u22121 = y+2 z\u22123 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (3; 2; \u22121). = (3; 2; \u22121). \u0110\u01b0\u1eddng th\u1eb3ng d : M 2 = \u22121 v\u1edbi d n\u00ean (P ) c\u00f3 vect\u01a1 ph\u00e1p tuy\u1ebfn #u\u00bb M\u1eb7t ph\u1eb3ng (P ) \u0111i 3 qua v\u00e0 vu\u00f4ng g\u00f3c V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) l\u00e0 3 (x \u2212 2) + 2 (y + 2) \u2212 (z \u2212 3) = 0 \u21d4 3x + 2y \u2212 z + 1 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 34 (C\u00e2u 23 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho ba \u0111i\u1ec3m A(0; \u22121; 3), B(1; 0; 1) v\u00e0 C(\u22121; 1; 2). Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh ch\u00ednh t\u1eafc c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng BC? B x \u2212 2y + z = 0. \uf8f1x = \u22122t \uf8f4 \uf8f2 A y = \u22121 + t \uf8f4 z = 3 + t. \uf8f3 C x y+1 z\u22123 D x\u22121 y z\u22121 \u22122 = = . == . 1 1 \u22122 1 1 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 B# C\u00bb (\u22122; 1; 1). V\u00ec \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm song song v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng BC n\u00ean ta ch\u1ecdn #u\u00bb (\u22122; 1; 1) l\u00e0m m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a n\u00f3. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh ch\u00ednh t\u1eafc c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 x y+1 z\u22123 \u22122 = = . 1 1 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 35 (C\u00e2u 35 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, vect\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua hai \u0111i\u1ec3m M (2; 3; \u22121) v\u00e0 N (4; 5; 3)? A u#\u00bb4 (1; 1; 1). B u#\u00bb3 (1; 1; 2). C u#\u00bb1 (3; 4; 1). D u#\u00bb2 (3; 4; 2). #\u00bb \u0253 L\u1eddi gi\u1ea3i. M N (2; 2; 4) Ta c\u00f3: = 2 (1; 1; 2) \u21d2 u#\u00bb3 (1; 1; 2)l\u00e0 m\u1ed9t v\u00e9c t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua hai \u0111i\u1ec3m M (2; 3; \u22121) v\u00e0 N (4; 5; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 36 (C\u00e2u 11 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). 33 3 Bi\u1ebft f (x) dx = 4 v\u00e0 g(x) dx = 1. Khi \u0111\u00f3 [f (x) \u2212 g(x)] dx b\u1eb1ng? 2 2 2 D 5. A \u22123. B 3. C 4. \u0253 L\u1eddi gi\u1ea3i. 3 33 Ta c\u00f3 [f (x) \u2212 g(x)] dx = f (x) dx \u2212 g(x) dx = 4 \u2212 1 = 3. 2 2 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 538 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 37 (C\u00e2u 9 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh ch\u00ednh t\u1eafc c\u1ee7a \uf8f1x = 1 + 2t \uf8f4 \uf8f2 \u0111\u01b0\u1eddng th\u1eb3ng d : y = 3t ? \uf8f4\uf8f3z = \u22122 + t A x+1 = y z\u22122 B x\u22121 y z+2 =. == \u22122 . 231 1 3 x+1 y z\u22122 x\u22121 y z+2 C 1 = 3 = \u22122 . D == . 231 \u0253 L\u1eddi gi\u1ea3i. D\u1ef1a v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 ta suy ra d qua A(1; 0; \u22122) v\u00e0 c\u00f3 VTCP #u\u00bb = (2; 3; 1) n\u00ean suy ra d c\u00f3 x\u22121 y z+2 ph\u01b0\u01a1ng tr\u00ecnh ch\u00ednh t\u1eafc l\u00e0 == 231 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 38 (C\u00e2u 33 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho c\u00e1c \u0111i\u1ec3m A(1; 2; 0), B(2; 0; 2), C(2; \u22121; 3), D(1; 1; 3). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua C v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABD) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = \u22122 \u2212 4t \uf8f1x = 2 + 4t \uf8f1x = \u22122 + 4t \uf8f1x = 4 + 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22122 \u2212 3t . B y = \u22121 + 3t . C y = \u22124 + 3t . D y=3\u2212t . \uf8f3\uf8f4z = 2 \u2212 t \uf8f3\uf8f4z = 3 \u2212 t \uf8f3\uf8f4z = 2 + t \uf8f3\uf8f4z = 1 + 3t #\u00bb \u0253 L\u1eddi gi\u1ea3i. AB #\u00bb \u00ee# \u00bb # \u00bb\u00f3 Ta c\u00f3 = (1; \u22122; 2), AD = (0; \u22121; 3) \u21d2 AB, AD = (\u22124; \u22123; \u22121). \u0110\u01b0\u1eddng th\u1eb3ng qua C(2; \u22121; 3) v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABD) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \uf8f1x = 2 + 4t \uf8f4 \uf8f2 y = \u22124 + 3t \uf8f4\uf8f3z = 2 + t. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 39 (C\u00e2u 32 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho c\u00e1c \u0111i\u1ec3m A(1; 0; 2), B(1; 2; 1), C(3; 2; 0) v\u00e0 D(1; 1; 3). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (BCD) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 1 \u2212 t \uf8f1x = 1 + t \uf8f1x = 2 + t \uf8f1x = 1 \u2212 t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = 4t . B y=4 . C y = 4 + 4t. D y = 2 \u2212 4t . \uf8f3\uf8f4z = 2 + 2t \uf8f4\uf8f3z = 2 + 2t \uf8f3\uf8f4z = 4 + 2t \uf8f4\uf8f3z = 2 \u2212 2t \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 B# C\u00bb = (2; 0; \u22121), B# D\u00bb = (0; \u22121; 2) v\u00e0 \u00eeB# C\u00bb, B# D\u00bb\u00f3 = (\u22121; \u22124; \u22122). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A\u00eeB#vC\u00e0\u00bb,vB#uD\u00f4\u00bbn\u00f3g=g\u00f3(c\u2212v1\u1edb; \u2212i m4;\u1eb7\u2212t 2p)hl\u1eb3\u00e0nvg\u00e9(cB-tC\u01a1 Dch)\u1ec9 th\u00ec vu\u00f4ng g\u00f3c v\u1edbi hai \u0111\u01b0\u1eddng th\u1eb3ng BC, BD n\u00ean nh\u1eadn v\u00e9c-t\u01a1 ph\u01b0\u01a1ng. C\u00f3 2 ph\u01b0\u01a1ng \u00e1n b\u1ecb lo\u1ea1i. Thay \u0111i\u1ec3m A(1; 0; 2) v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1ed9t trong hai ph\u01b0\u01a1ng \u00e1n c\u00f2n l\u1ea1i, \uf8f1x = 2 + t \uf8f11 = 2 + t \uf8f1t = \u22121 \uf8f4 \uf8f4\uf8f4 \uf8f2 \uf8f2\uf8f2 ch\u1eb3ng h\u1ea1n thay v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh y = 4 + 4t ta \u0111\u01b0\u1ee3c 0 = 4 + 4t \u21d4 t = \u22121 (th\u1ecfa m\u00e3n). \uf8f3\uf8f4z = 4 + 2t \uf8f4\uf8f32 = 4 + 2t \uf8f3\uf8f4t = \u22121 \uf8f1x = 2 + t \uf8f4 \uf8f2 V\u1eady \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (BCD) l\u00e0 y = 4 + 4t \uf8f4\uf8f3z = 4 + 2t Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 539 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 40 (C\u00e2u 31 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz cho A(0; 0; 2), B(2; 1; 0), C(1; 2; \u22121) v\u00e0 D(2; 0; \u22122). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (BCD) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 3 + 3t \uf8f1x = 3 . \uf8f1x = 3 + 3t \uf8f1x = 3t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22122 + 2t . B y=2 C y = 2 + 2t . D y = 2t . \uf8f4\uf8f3z = 1 \u2212 t \uf8f3\uf8f4z = \u22121 + 2t \uf8f3\uf8f4z = 1 \u2212 t \uf8f4\uf8f3z = 2 + t \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi d lB#\u00e0C\u00bb\u0111\u01b0=\u1eddn(g\u2212t1h; \u1eb31n; \u2212g 1\u0111)i; qB#uDa\u00bb A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (BCD). Ta c\u00f3 = (0; \u22121; \u22122). #n\u00bb(BCD) \u00eeB# D\u00bb, B# C\u00bb\u00f3 M\u1eb7t ph\u1eb3ng (BCD) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 = = (3; 2; \u22121). G\u1ecdi #u\u00bbd l\u00e0 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d. V\u00ec d \u22a5 (B CD) n\u00ean u#\u00bbd = #n\u00bb(BCD) = (3; 2; \u22121). \uf8f1x = 3 \uf8f1x = 3t \uf8f4 \uf8f4 \uf8f2 \uf8f2 \u0110\u00e1p \u00e1n y = 2 v\u00e0 y = 2t c\u00f3 vec-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng kh\u00f4ng c\u00f9ng ph\u01b0\u01a1ng v\u1edbi vec-t\u01a1 \uf8f4\uf8f3z = \u22121 + 2t \uf8f4\uf8f3z = 2 + t u#\u00bbd = (3; 2; \u22121) n\u00ean lo\u1ea1i. \uf8f1x = 3 + 3t \uf8f1x = 3 + 3t \uf8f4 \uf8f4 \uf8f2\uf8f2 Ta th\u1ea5y \u0111i\u1ec3m A(0; 0; 2) kh\u00f4ng th\u1ecfa h\u1ec7 y = \u22122 + 2t n\u00ean lo\u1ea1i \u0111\u00e1p \u00e1n y = \u22122 + 2t . \uf8f3\uf8f4z = 1 \u2212 t \uf8f4\uf8f3z = 1 \u2212 t Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 41 (C\u00e2u 38 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec1m M (1; 0; 1) v\u00e0 N (3; 2; \u22121). \u0110\u01b0\u1eddng th\u1eb3ng M N c\u00f3 ph\u01b0\u01a1ng tham s\u1ed1 l\u00e0 \uf8f1x = 1 \u2212 2t \uf8f1x = 1 \u2212 t \uf8f1x = 1 + t \uf8f1x = 1 + 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 B y = 2t . C y=t . D y=t . A y = 2t . \uf8f4\uf8f3z = 1 + t \uf8f4\uf8f3z = 1 + t \uf8f3\uf8f4z = 1 + t \uf8f3\uf8f4z = 1 \u2212 t \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 M# N\u00bb = (2; 2; \u22122) = 2(1; 1; \u22121). \u0110\u01b0\u1eddng th\u1eb3ng M N \u0111i qua \u0111i\u1ec3m M (1; 0; 1) v\u00e0 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (1; 1; \u22121) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 l\u00e0 \uf8f1x = 1 + t \uf8f4 \uf8f2 y=t \uf8f4\uf8f3z = 1 \u2212 t. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 42 (C\u00e2u 32 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A(1; 0; 1), B(1; 1; 0) v\u00e0 C(3; 4; \u22121). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi BC c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22121 y z\u22121 A B x+1 y z+1 == . == . 4 5 \u22121 2 3 \u22121 x\u22121 y z\u22121 x+1 y z+1 C == \u22121 . D == \u22121 . 2 3 4 5 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 540 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN Ta c\u00f3 B# C\u00bb = (2; 3; \u22121). 1) v\u00e0 nh\u1eadn B# C\u00bb = (2; 3; \u22121) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A(1; 0; x\u22121 y z\u22121 == \u22121 . 2 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 43 (C\u00e2u 35 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz cho ba \u0111i\u1ec3m A(1; 2; 3), B(1; 1; 1), C(3; 4; 0) \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi BC c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22121 y\u22122 z\u22123 = = .. A x+1 y+2 z+3 B = =. 451 451 x\u22121 y\u22122 z\u22123 x+1 y+2 z+3 C == \u22121 . D 2 = 3 = \u22121 . 23 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 #\u00bb = (2; 3; \u22121) . #\u00bb BC BC Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A(1; 2; 3) nh\u1eadn = (2; 3; \u22121) l\u00e0 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 d\u1ea1ng x\u22121 y\u22122 z\u22123 2 = 3 = \u22121 . Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 44 (C\u00e2u 34 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A (1; 2; 0), B (1; 1; 2) v\u00e0 C (2; 3; 1). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A (1; 2; 0) v\u00e0 song song v\u1edbi BC c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x\u22121 y\u22122 z B x\u22121 y\u22122 z 1 = 2 = \u22121. = =. 3 43 x+1 y+2 z x+1 y+2 z C = = . D 1 = 2 = \u22121. 3 43 \u0253 L\u1eddi gi\u1ea3i. B# C\u00bb = (1; 2; \u22121). A (1; 2; 0) v\u00e0 song song v\u1edbi BC nh\u1eadn B# C\u00bb = (1; 2; \u22121) l\u00e0m vecto ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua x\u22121 y\u22122 z ph\u01b0\u01a1ng tr\u00ecnh ch\u00ednh t\u1eafc l\u00e0: 1 = 2 = \u22121. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 45 (C\u00e2u 35 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A (1; 1; 0) ; B (1; 0; 1) ; C (3; 1; 0). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A (1; 1; 0) v\u00e0 song song v\u1edbi BC c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh A x+1 y+1 z B x+1 y+1 z 2 = 1 = \u22121 . = =. 4 11 x\u22121 y\u22121 z x\u22121 y\u22121 z C == \u22121 . D = =. 4 11 21 \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm \u0111i qua A (1; 1; 0) v\u00e0 c\u00f3 m\u1ed9t vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = B# C\u00bb = (2; 1; \u22121). x\u22121 y\u22121 z Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 2 = 1 = \u22121. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 541 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian \u0104 C\u00e2u 46 (C\u00e2u 35 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (1; \u22122; 3) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 y + 3z + 1 = 0. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) l\u00e0 \uf8f1x = 1 + 2t \uf8f1x = \u22121 + 2t \uf8f1x = 2 + t \uf8f1x = 1 \u2212 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22122 \u2212 t . B y = 2 \u2212 t . C y = \u22121 \u2212 2t . D y = \u22122 \u2212 t . \uf8f4\uf8f3z = 3 + 3t \uf8f4\uf8f3z = \u22123 + 3t \uf8f4\uf8f3z = 3 + 3t \uf8f4\uf8f3z = 3 \u2212 3t \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi d l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm. d vu\u00f4ng g\u00f3c v\u1edbi (P ) n\u00ean VTCP c\u1ee7a d l\u00e0 #u\u00bbd = #n\u00bbP = (2; \u22121; 3). \uf8f1x = 1 + 2t \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 y = \u22122 \u2212 t \uf8f3\uf8f4z = 3 + 3t. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 47 (C\u00e2u 38 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (1; 2; \u22123) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 y + 3z \u2212 1 = 0. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) l\u00e0 \uf8f1x = 2 + t \uf8f1x = \u22121 + 2t \uf8f1x = 1 + 2t \uf8f1x = 1 \u2212 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22121 + 2t . B y = \u22122 \u2212 t . C y=2\u2212t . D y=2\u2212t . \uf8f4\uf8f3z = 3 \u2212 3t \uf8f3\uf8f4z = 3 + 3t \uf8f4\uf8f3z = \u22123 + 3t \uf8f4\uf8f3z = \u22123 \u2212 3t \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng(P ) nh\u1eadn #n\u00bb = (2; \u22121; 3) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng. \uf8f1x = 1 + 2t \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 y = 2 \u2212 t \uf8f3\uf8f4z = \u22123 + 3t. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 48 (C\u00e2u 34 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (1; \u22122; 2) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x + y \u2212 3z + 1 = 0. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) l\u00e0 \uf8f1x = 1 + 2t \uf8f1x = 1 + t \uf8f1x = 2 + t \uf8f1x = \u22121 + 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22122 + t . B y = \u22122 \u2212 2t . C y = 1 \u2212 2t . D y=2+t . \uf8f3\uf8f4z = 2 \u2212 3t \uf8f4\uf8f3z = 2 + t \uf8f4\uf8f3z = \u22123 + 2t \uf8f3\uf8f4z = \u22122 \u2212 3t \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi d l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm. d vu\u00f4ng g\u00f3c v\u1edbi (P ) n\u00ean VTCP c\u1ee7a d l\u00e0 #u\u00bbd = #n\u00bbP = (2; 1; \u22123). \uf8f1x = 1 + 2t \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 y = \u22122 + t \uf8f3\uf8f4z = 2 \u2212 3t. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 49 (C\u00e2u 32 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (\u22121; 3; 2) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y + 4x + 1 = 0. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 542 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN A x+1 y\u22123 z\u22122 B x\u22121 y+3 z+2 = \u22122 = . = \u22122 = . 1 1 1 1 x\u22121 y+3 z+2 x+1 y\u22123 z\u22122 C 1 = \u22122 = . D = \u22122 = . 1 4 4 \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua M (\u22121; 3; 2) v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = #n\u00bbP = (1; \u22122; 4). x+1 y\u22123 z\u22122 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 = = . 1 \u22122 4 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 50 (C\u00e2u 34 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; 1; \u22121) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x \u2212 3y + 2z + 1 = 0. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x\u22122 y\u22121 z+1 B x\u22122 y\u22121 z+1 = \u22123 = . = \u22123 = . 1 1 1 2 x+2 y+1 z\u22121 x+2 y+1 z\u22121 C 1 = \u22123 =. D = \u22123 = . 1 2 1 \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (1; \u22123; 2). \u0110\u01b0\u1eddng th\u1eb3ng d vu\u00f4ng g\u00f3c v\u1edbi (P ) n\u00ean nh\u1eadn v\u00e9c-t\u01a1 #n\u00bb l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua M (2; 1; \u22121) v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (P ) l\u00e0 x\u22122 y\u22121 z+1 = \u22123 = . 1 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 51 (C\u00e2u 29 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (1; 2; \u22121) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x + y \u2212 3z + 1 = 0. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x\u22121 y\u22122 z+1 B x\u22121 y\u22122 z+1 = =. 2 = 1 = \u22123 . 211 x+1 y+2 z\u22121 x+1 y+2 z\u22121 C = =. D 2 = 1 = \u22123 . 211 \u0253 L\u1eddi gi\u1ea3i. (P ) c\u00f3 #n\u00bb = (2; 1; \u22123) l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn. \u0110\u01b0\u1eddng th\u1eb3ng d vu\u00f4ng g\u00f3c v\u1edbi (P ) n\u00ean #n\u00bb c\u0169ng l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d qua M (1; 2; \u22121) v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) l\u00e0 x\u22121 y\u22122 z+1 2 = 1 = \u22123 . Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 52 (C\u00e2u 29 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; 1; \u22122) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 3x + 2y \u2212 z + 1 = 0. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh A x\u22122 y\u22121 z+2 B x\u22122 y\u22121 z+2 3 = 2 = \u22121 . = =. 321 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 543 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian C x+2 y+1 z\u22122 D x+2 y+1 z\u22122 = =. 3 = 2 = \u22121 . 321 \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn #n\u00bb = (3; 2; \u22121). G\u1ecdi d l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh. Do d \u22a5 (P ) n\u00ean d c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #n\u00bb = (3; 2; \u22121). x\u22122 y\u22121 z+2 Ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d : 3 = 2 = \u22121 . Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 53 (C\u00e2u 36 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m M (1; 1; \u22121) v\u00e0 N (3; 0; 2). \u0110\u01b0\u1eddng th\u1eb3ng M N c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22121 y\u22121 z+1 x+1 y+1 z\u22121 A = =. B 2 = \u22121 = . 411 3 x\u22121 y\u22121 z+1 x+1 y+1 z\u22121 C = =. D = \u22121 = . 411 2 3 \u0110\u01b0\u1eddng th\u1eb3ng M N c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 \u0253 LM#\u1eddNi\u00bbg=i\u1ea3i(.2; M\u2212# 1N;\u00bb3)l\u00e0. m Do \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng ph\u01b0\u01a1ng l\u00e0 M N \u0111i qua M v\u00e0 nh\u1eadn v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 x\u22121 y\u22121 z+1 = \u22121 = . 2 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 54 (C\u00e2u 29 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m M (1; 0; 1) v\u00e0 N (4; 2; \u22122). \u0110\u01b0\u1eddng th\u1eb3ng M N c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22121 y z\u22121 x\u22121 y z\u22121 A == \u22123 . B == \u22121 . 3 2 5 2 x+1 y z+1 x+1 y z+1 C == \u22121 . D == \u22123 . 5 2 3 2 \u0110\u01b0\u1eddng th\u1eb3ng M N \u0111i qua \u0111i\u1ec3m M (1; 0; 1) v\u00e0\u0253nLh\u1edd\u1eadni gM#i\u1ea3Ni.\u00bb = (3; 2; \u22123) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng n\u00ean c\u00f3 x\u22121 y z\u22121 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 == \u22123 . 3 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 55 (C\u00e2u 29 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m M (1; 1; 0) v\u00e0 N (3; 2; \u22121). \u0110\u01b0\u1eddng th\u1eb3ng M N c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22121 y\u22121 z x+1 y+1 z 4 = 3 = \u22121. A 4 = 3 = \u22121. B C x\u22121 y\u22121 z D x+1 y+1 z 2 = 1 = \u22121 . 2 = 1 = \u22121 . \u0110\u01b0\u1eddng th\u1eb3ng MN nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = #\u00bb = \u0253 L\u1eddi gi\u1ea3i. vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng v\u00e0 \u0111i qua \u0111i\u1ec3m M (1; 1; 0) MN (2; 1; \u22121) l\u00e0m n\u00ean M N c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22121 y\u22121 z 2 = 1 = \u22121. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 544 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 56 (C\u00e2u 29 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A(1; 2; \u22121), B(3; 0; 1) v\u00e0 C(2; 2; \u22122). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x\u22121 y\u22122 z+1 B x+1 y+2 z\u22121 = \u22122 = . = =. 1 3 121 x\u22121 y\u22122 z\u22121 x\u22121 y\u22122 z+1 C == \u22121 . D = =. 121 12 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 A# B\u00bb = (2; \u22122; 2), A# C\u00bb = (1; 0; \u22121). Suy ra m\u1eb7t ph\u1eb3ng (ABC) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb(ABC) = [A# B\u00bb, A# C\u00bb] = (1; 2; 1). #n\u00bb(ABC) = (2; 4; 2). \u0110\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) n\u00ean c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = Ta ch\u1ecdn VTCP l\u00e0 #u\u00bb = (1; 2; 1) V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 x\u22121 = y\u22122 = z+1 \u00b7 1 2 1 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 57 (C\u00e2u 34 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A(1; 2; \u22121), B(3; 0; 1), C(2; 2; \u22122). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x\u22121 y\u22122 z\u22121 B x\u22121 y\u22122 z+1 1 = 2 = \u22121 . = \u22122 = . 1 3 x\u22121 y\u22122 z+1 x+1 y+2 z\u22121 C == . D = =. 121 121 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 A# B\u00bb = (2; \u22122; 2), A# C\u00bb = (1; 0; \u22121). Suy ra \u00eeA# B\u00bb, A# C\u00bb\u00f3 = (2; 4; 2) = 2 #u\u00bb, #u\u00bb = (1; 2; 1). \u0110\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) n\u00ean \u0111\u01b0\u1eddng th\u1eb3ng nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb = (1; 2; 1) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng v\u00e0 do \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A(1; 2; \u22121) n\u00ean ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 x \u2212 1 = y \u2212 2 = z + 1\u00b7 121 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 58 (C\u00e2u 37 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; \u22122; 1) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 3y \u2212 z + 1 = 0. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1 x = 2 + 2t \uf8f1 x = 2 + 2t \uf8f1 x = 2 + 2t \uf8f1 x = 2 + 2t \uf8f2 \uf8f2\uf8f2 \uf8f2 A y = 2 \u2212 3t . D y = \u22123 \u2212 2t . B y = \u22122 \u2212 3t . C y = \u22122 + 3t . \uf8f3 z = 1 \u2212 t. \uf8f3 z = 1 \u2212 t. \uf8f3 z = 1 + t. \uf8f3 z = \u22121 + t. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi d l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ). Do d vu\u00f4ng g\u00f3c v\u1edbi (P ) n\u00ean d c\u00f3 m\u1ed9t vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = (2; \u22123; \u22121). \uf8f1 x = 2 + 2t \uf8f2 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 y = \u22122 \u2212 3t \uf8f3 z = 1 \u2212 t. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 59 (C\u00e2u 38 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho ba \u0111i\u1ec3m A(2; \u22122; 3), B(1; 3; 4) v\u00e0 C(3; \u22121; 5). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 song song v\u1edbi BC c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0: Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 545 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian A x\u22122 y+4 z\u22121 B x+2 y\u22122 z+3 = \u22122 = . = \u22124 = . 2 3 2 1 x\u22122 y+2 z\u22123 x\u22122 y+2 z\u22123 C = =. D = \u22124 = . 429 2 1 \u0253 L\u1eddi gi\u1ea3i. B# C\u00bb = (2; \u22124; 1). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A song song v\u1edbi BC n\u00ean nh\u1eadn B# C\u00bb l\u00e0m m\u1ed9t v\u00e9ct\u01a1 ch\u1ec9 ph\u01b0\u01a1ng. x\u22122 y+2 z\u22123 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 = \u22124 = . 2 1 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 60 (C\u00e2u 16 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). x+1 y\u22122 z\u22121 Trong kh\u00f4ng gian Oxyz, \u0111i\u1ec3m n\u00e0\u1ecd d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng d : \u22121 = = ? 3 3 A P (\u22121; 2; 1). B Q (1; \u22122; \u22121). C N (\u22121; 3; 2). D M (1; 2; 1). \u0253 L\u1eddi gi\u1ea3i. C\u00e1ch 1 : V\u00ec ph\u01b0\u01a1ng tr\u00ecnh ch\u00ednh t\u1eafc c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua P (xo; yo; z0)v\u00e0 c\u00f3 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb (a; b; c) l\u00e0 x \u2212 xo = y \u2212 yo = z \u2212 zo n\u00ean d\u1ec5 d\u00e0ng th\u1ea5y \u0111i\u1ec3m P (\u22121; 2; 1)thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng d . abc C\u00e1ch 2 : Thay t\u1ecda \u0111\u1ed9 4 \u0111i\u1ec3m M ,N ,P ,Qv\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d ta th\u1ea5y \u0111i\u1ec3m P th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 61 (C\u00e2u 13 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d: x\u22122 = y\u22121 = z+1 \u00b7 \u0110i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y thu\u1ed9c d? 1 \u22122 3 A P (2; 1; \u22121). B M (1; 2; 3). C Q(2; 1; 1). D N (1; \u22122; 3). \u0253 L\u1eddi gi\u1ea3i. Thay t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m P (2; 1; \u22121) v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng (d) ta c\u00f3: 2\u22122 = 1\u22121 = \u22121+1 \u21d4 0 = 0 = 0 = 0 (th\u1ecfa m\u00e3n). 1 \u22122 3 1 \u22122 3 Thay t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m M (1; 2; 3) v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng (d) ta c\u00f3: 1\u22122 = 2\u22121 = 3+1 \u21d4 \u22121 = 1 = 4 (v\u00f4 l\u00ed). 1 \u22122 3 1 \u22122 3 Thay t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m Q(2; 1; 1) v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng (d) ta c\u00f3: 2\u22122 = 1\u22121 = 1+1 \u21d4 0 = 0 = 2 (v\u00f4 l\u00ed). 1 \u22122 3 1 \u22122 3 Thay t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m N (1; \u22122; 3) v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng (d) ta c\u00f3: 1\u22122 = \u22122\u22121 = 3+1 \u21d4 \u22121 = \u22123 = 4 (v\u00f4 l\u00ed). 1 \u22122 3 1 \u22122 3 V\u1eady \u0111i\u1ec3m P (2; 1; \u22121) thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng (d). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 62 (C\u00e2u 30 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 2y \u2212 z + 1 = 0 v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng x\u22121 y+2 z\u22121 \u2206 : = = . T\u00ednh kho\u1ea3ng c\u00e1ch d gi\u1eefa \u2206 v\u00e0 (P ). 212 1 5 2 A d= . B d= . C d= . D d = 2. 3 3 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng \u2206 \u0111i qua \u0111i\u1ec3m M (1; \u22122; 1) v\u00e0 c\u00f3 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (2; 1; 2). M\u1eb7t ph\u1eb3ng (P ) c\u00f3 vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (2; \u22122; \u22121). Ta c\u00f3 #u\u00bb. #n\u00bb = 2.2 + 1.(\u22122) + 2.(\u22121) = 0. Th\u1ebf t\u1ecda \u0111\u1ed9 M (1; \u22122; 1) v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) ta c\u00f3 2 + 4 \u2212 1 + 1 = 0 ( v\u00f4 l\u00fd). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 546 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN V\u1eady \u2206 \u2225 (P ). Suy ra d (\u2206, (P )) = d (M, (P )) = |2.1 \u2212 2.(\u22122) \u2212 1 + 1| = 2. 22 + (\u22122)2 + (\u22121)2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 63 (C\u00e2u 47 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). x+1 y z\u22125 Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : 1 = \u22123 = \u22121 v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 3x \u2212 3y + 2z + 6 = 0. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang ? A d c\u1eaft v\u00e0 kh\u00f4ng vu\u00f4ng g\u00f3c v\u1edbi (P ). B d vu\u00f4ng g\u00f3c v\u1edbi (P ). C d song song v\u1edbi (P ). D d n\u1eb1m trong (P ). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua M (\u22121; 0; 5) c\u00f3 vtcp #u\u00bb = (1; \u22123; \u22121) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) c\u00f3 vtpt #n\u00bb = (3; \u22123; 2). M \u2208\/ P \u21d2 lo\u1ea1i \u0111\u00e1p \u00e1n D. #n\u00bb, #u\u00bb kh\u00f4ng c\u00f9ng ph\u01b0\u01a1ng \u21d2 lo\u1ea1i \u0111\u00e1p \u00e1n B. #n\u00bb. #u\u00bb = 10 \u21d2 #n\u00bb, #u\u00bb kh\u00f4ng vu\u00f4ng g\u00f3c \u21d2 lo\u1ea1i \u0111\u00e1p \u00e1n C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 64 (C\u00e2u 31 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (2; \u22121; 2) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d: x\u22121 = y +2 = z\u22123 M\u1eb7t . 231 ph\u1eb3ng \u0111i qua \u0111i\u1ec3m qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A 2x + 3y + z \u2212 3 = 0. B 2x \u2212 y + 2z \u2212 9 = 0. C 2x + 3y + z + 3 = 0. D SAM . \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 (P ) \u22a5 d \u21d2 vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) l\u00e0 #n\u00bb = #u\u00bb = (2; 3; 1). Khi \u0111\u00f3 m\u1eb7t ph\u1eb3ng (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh 2x + 3y + z \u2212 3 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 65 (C\u00e2u 19 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(3; 4; 1) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy)? A Q(0; 4; 1). B P (3; 0; 1). C M (0; 0; 1). D N (3; 4; 0). \u0253 L\u1eddi gi\u1ea3i. #\u00bb Ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 z = 0 v\u00e0 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 k = (0; 0; 1). \uf8f1x = 3 \uf8f4 \uf8f2 G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (Oxy) khi \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u2206 : y = 4 \uf8f4\uf8f3z = 1 + t. Do \u0111\u00f3 t\u1ecda \u0111\u1ed9 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a M l\u00ean m\u1eb7t ph\u1eb3ng (Oxy) th\u1ecfa m\u00e3n h\u1ec7 \uf8f1z = 0 \uf8f1x = 3 \uf8f4\uf8f4 \uf8f4\uf8f4 \uf8f2\uf8f4x = 3 \u21d4 \uf8f2\uf8f4y = 4 \uf8f4y = 4 \uf8f4z = 0 \uf8f4\uf8f4 \uf8f4 \uf8f4 = \u22121. \uf8f3z = 1 + t \uf8f3t Suy ra \u0111i\u1ec3m c\u1ea7n t\u00ecm l\u00e0 N (3; 4; 0). 547 S\u0110T: 0905.193.688 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t"]
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