Toàn cảnh đề thi tốt nghiệp THPT môn Toán (2018 – 2022)

["3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian \u0104 C\u00e2u 66 (C\u00e2u 21 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, m\u1eb7t ph\u1eb3ng \u0111i qua \u0111i\u1ec3m A(1; 2; \u22122) v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng x+1 y\u22122 z+3 \u2206 : = = c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 213 A 3x + 2y + z \u2212 5 = 0. B 2x + y + 3z + 2 = 0. C x + 2y + 3z + 1 = 0. D 2x + y + 3z \u2212 2 = 0. M\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u2206 l\u00e0\u0253#u\u00bbL\u1edd=i gi\u1ea3i. (2; 1; 3). V\u00ec m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng \u2206 n\u00ean c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = #u\u00bb = (2; 1; 3). Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 2(x \u2212 1) + 1(y \u2212 2) + 3(z + 2) = 0 \u21d4 2x + y + 3z + 2 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 67 (C\u00e2u 29 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m M (1; 1; \u22122) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d: x\u22121 = y+2 = z M\u1eb7t 1 2 \u22123 . 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 3z \u2212 9 = 0. B x + y \u2212 2z \u2212 6 = 0. C x + 2y \u2212 3z + 9 = 0. D x + y \u2212 2z + 6 = 0. M\u1eb7t ph\u1eb3ng \u0111i qua M v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d \u0253 L\u1eddi gi\u1ea3i. ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (1; 2; \u22123). n\u00ean nh\u1eadn m\u1ed9t v\u00e9c-t\u01a1 Suy ra m\u1eb7t ph\u1eb3ng \u0111i qua \u0111i\u1ec3m M n\u00ean c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 1 (x \u2212 1) + 2 (y \u2212 1) \u2212 3 (z + 2) = 0 \u21d4 x + 2y \u2212 3z \u2212 9 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 68 (C\u00e2u 34 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m A(1; \u22122; 3) v\u00e0 hai m\u1eb7t ph\u1eb3ng (P ) : x+y+z+1 = 0, (Q) : x \u2212 y + z \u2212 2 = 0. Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A, song song v\u1edbi (P ) v\u00e0 (Q)? \uf8f1x = 1 \uf8f1x = 1 + 2t \uf8f1x = 1 + t \uf8f4 \uf8f4 \uf8f4 \uf8f1x = \u22121 + t \uf8f2 \uf8f2 \uf8f2 \uf8f4 \uf8f2 B y = \u22122 C y = \u22122 D y = \u22122 A y=2 \uf8f3\uf8f4z = \u22123 \u2212 t. \uf8f3\uf8f4z = 3 \u2212 2t. \uf8f3\uf8f4z = 3 + 2t. \uf8f4 z = 3 \u2212 t. \uf8f3 \u0253 L\u1eddi gi\u1ea3i. ph\u00e1p tuy\u1ebfn n#\u00bb1(1; 1; 1), (Q) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn n#\u00bb2(1; \u22121; 1). (P ) c\u00f3 v\u00e9c-t\u01a1 = (2; 0; \u22122). v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng Ta c\u00f3 [n#\u00bb1, n#\u00bb2] c\u1ea7n t\u00ecm nh\u1eadn v\u00e9c-t\u01a1 #u\u00bb(1; 0; \u22121) l\u00e0m \u0110\u01b0\u1eddng th\u1eb3ng \uf8f1x = 1 + t \uf8f4 \uf8f2 c\u1ea7n t\u00ecm l\u00e0 y = \u22122 \uf8f4\uf8f3z = 3 \u2212 t. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 69 (C\u00e2u 3 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111i\u1ec3m A (1; 1; 0) v\u00e0 B (0; 1; 2). V\u00e9ct\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t #b\u00bbv\u00e9=ct\u01a1(\u2212ch1;\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng AB? C #d\u00bb = (\u22121; 1; 2). D #a\u00bb = (\u22121; 0; \u22122). A 0; 2). B #c\u00bb = (1; 2; 2). A# B\u00bb = (\u22121; 0; 2) l\u00e0 m\u1ed9t v\u00e9ct\u01a1 ch\u1ec9 ph\u01b0\u01a1ng \u0253 L\u1eddi gi\u1ea3i. AB. c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 548 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 70 (C\u00e2u 15 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m M (1; 2; 3). G\u1ecdi M1, M2 l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a M tr\u00ean c\u00e1c tr\u1ee5c Ox, Oy. V\u00e9ct\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 v\u00e9ct\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng M1M2? B u#\u00bb3 = (1; 0; 0). C u#\u00bb4 = (\u22121; 2; 0). D u#\u00bb1 = (0; 2; 0). A u#\u00bb2 = (1; 2; 0). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 M1 (1; 0; 0) v\u00e0 M2 (0; 2; 0). Do \u0111\u00f3, #\u00bb = (\u22121; 2; 0) l\u00e0 m\u1ed9t v\u00e9ct\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng M1M2 M1M2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 71 (C\u00e2u 49 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m A(1; 0; 2) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh: x\u22121 y z+1 . Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u2206 \u0111i qua A, vu\u00f4ng g\u00f3c v\u00e0 c\u1eaft d. == 112 x\u22121 y z+2 x\u22121 y z+2 A \u2206: = = . B \u2206: == . 111 1 1 \u22121 x\u22121 y z\u22122 x\u22121 y z\u22122 C \u2206: = = . D \u2206: 1 = \u22123 =. 221 1 \u0253 L\u1eddi gi\u1ea3i. C\u00e1ch 1 : ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 (P ) : x + y + 2z \u2212 5 = 0. giao \u0111i\u1ec3m c\u1ee7a d v\u00e0 (P ) l\u00e0 B(2; 1; 1). khi \u0111\u00f3 \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm ch\u00ednh l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 B c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh x\u22121 = y z+2 1 = \u22121 1 C\u00e1ch 2 : G\u1ecdi B(1 + b; b; \u22121 + 2b) l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u2206 v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng d. ta c\u00f3 \u2206 vu\u00f4ng g\u00f3c v\u1edbi d n\u00ean A# B\u00bb.u# \u2206\u00bb = 0 hay b + b + 2(2b \u2212 3) = 0 suy ra b = 1 v\u00e0 B(2; 1; 1). khi \u0111\u00f3 \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm ch\u00ednh l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A v\u00e0 B c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh x\u22121 = y z+2 1 = \u22121 1 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 72 (C\u00e2u 37 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). x\u22121 y+5 z\u22123 Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : = \u22121 = . Ph\u01b0\u01a1ng tr\u00ecnh 2 4 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng h\u00ecnh h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean m\u1eb7t ph\u1eb3ng x + 3 = 0 ? \uf8f1x = \u22123 \uf8f1x = \u22123 \uf8f1x = \u22123 \uf8f1x = \u22123 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22125 \u2212 t . B y = \u22125 + t . C y = \u22125 + 2t. D y = \u22126 \u2212 t . \uf8f3\uf8f4z = \u22123 + 4t \uf8f3\uf8f4z = 3 + 4t \uf8f3\uf8f4z = 3 \u2212 t \uf8f3\uf8f4z = 7 + 4t Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 549","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian C\u00e1ch 1: \u0110\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m M0(1; \u22125; 3) v\u00e0 c\u00f3 VTCP #u\u00bbd = (2; \u22121; 4) G\u1ecdi (Q) l\u00e0 m\u1eb7t ph\u1eb3ng ch\u1ee9a d v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) : x + 3 = 0. Suy ra m\u1eb7t ph\u1eb3ng (Q) \u0111i qua \u0111i\u1ec3m M0(1; \u22125; 3) v\u00e0 c\u00f3 VTPT l\u00e0 [ #n\u00bbP ; #u\u00bbd] = (0; 4; 1) \u21d2 (Q) : 4y + z + 17 = 0. Ph\u01b0\u01a1ng tr\u00ecnh h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean m\u1eb7t ph\u1eb3ng (P ) l\u00e0 \u00ae4y + z + 17 = 0 \uf8f1x = \u22123 \uf8f4 \uf8f2 x+3=0 hay y = \u22126 \u2212 t. \uf8f4\uf8f3z = 7 + 4t C\u00e1ch 2. Tr\u1eafc nghi\u1ec7m. G\u1ecdi I = d \u2229 (\u03b1), suy ra I(\u22123; \u22123; \u22125). D\u1ec5 th\u1ea5y ch\u1ec9 c\u00f3 \u0111\u00e1p \u00e1n D th\u1ecfa m\u00e3n Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 73 (C\u00e2u 33 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). x\u22123 y\u22121 z+7 Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; 3) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : 2 = 1 = \u22122 . \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A, vu\u00f4ng g\u00f3c v\u1edbi d v\u00e0 c\u1eaft tr\u1ee5c Ox c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = \u22121 + 2t \uf8f1x = 1 + t \uf8f1x = \u22121 + 2t \uf8f1x = 1 + t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 B y = 2 + 2t. C y = \u22122t . D y = 2 + 2t. A y = 2t . \uf8f3\uf8f4z = 3t \uf8f4\uf8f3z = 3 + 2t \uf8f4\uf8f3z = t \uf8f4\uf8f3z = 3 + 3t \u0253 L\u1eddi gi\u1ea3i. GDo\u1ecdi\u2206\u2206\u22a5l\u00e0d\u0111, \u01b0\u2206\u1eddnqguathA\u1eb3nng\u00eacn\u1ea7nB# At\u00bb\u00ecm\u00b7 u#v\u00bbd\u00e0=B0=\u21d4\u22062\u2229(1O\u2212xb\u21d2) +B2(\u2212b; 0; 0) v\u00e0 B# A\u00bb = (1 \u2212 b; 2; 3). 6 = 0 \u21d4 b = \u22121. \uf8f1x = \u22121 + 2t B# A\u00bb = (2; 2; 3) n\u00ean \u2206 : \uf8f4 T\u1eeb \u0111\u00f3 \u2206 qua B(\u22121; 0; 0), c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 \uf8f2 y = 2t \uf8f4\uf8f3z = 3t. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 74 (C\u00e2u 49 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). \uf8f1x = 1 + 3t \uf8f4 \uf8f2 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : y = 1 + 4t . G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m \uf8f3\uf8f4z = 1 A(1; 1; 1) v\u00e0 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (1; \u22122; 2). \u0110\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c nh\u1ecdn t\u1ea1o b\u1edfi d v\u00e0 \u2206 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = \u22121 + 2t \uf8f1x = \u22121 + 2t \uf8f1x = 1 + 3t \uf8f1x = 1 + 7t \uf8f4\uf8f4\uf8f4 \uf8f4 \uf8f2 \uf8f2\uf8f2\uf8f2 A y=1+t . B y = \u221210 + 11t. C y = \u221210 + 11t . D y = 1 + 4t . \uf8f4\uf8f3z = 1 + 5t \uf8f3\uf8f4z = \u22126 \u2212 5t \uf8f3\uf8f4z = 6 \u2212 5t \uf8f3\uf8f4z = 1 \u2212 5t \u0253 L\u1eddi gi\u1ea3i. \uf8f1x = 1 + t \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 \u0111\u01b0\u1eddng th\u1eb3ng \u2206 : y = 1 \u2212 2t cA#\u00f3C\u00bbA# =B\u00bb \uf8f3\uf8f4z = 1 + 2t . Ch\u1ecdn \u0111i\u1ec3m B(0; 3; \u22121) \u2208 \u2206 ta = (\u22121; 2; \u22122) v\u00e0 AB = 3. Ch\u1ecdn \u0111i\u1ec3m C(4; 5; 1) \u2208 d ta c\u00f3 (3; 4; 0) v\u00e0 AC = 5. Ta c\u00f3 A# B\u00bb \u00b7 A# C\u00bb = 5 > 0 \u21d2 B\u2019AC < 90\u25e6. Ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c nh\u1ecdn B\u2019AC c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = AC \u00b7 A# B\u00bb + AB \u00b7 A# C\u00bb = (4; 22; \u221210). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 550 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0110\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c nh\u1ecdn t\u1ea1o b\u1edfi d v\u00e0 \u2206 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f9ng ph\u01b0\u01a1ng v\u1edbi v\u00e9c- \uf8f1x = \u22121 + 2t A# C\u00bb \uf8f4 t\u01a1 = (4; 22; \u221210). X\u00e9t ph\u01b0\u01a1ng \u00e1n \uf8f2 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #v\u00bb = (2; 11; \u22125) c\u00f9ng y = \u221210 + 11t \uf8f4\uf8f3z = 6 \u2212 5t. ph\u01b0\u01a1ng v\u1edbi v\u00e9c-t\u01a1 A# C\u00bb = (4; 22; \u221210) v\u00e0 \u0111i qua \u0111i qua \u0111i\u1ec3m A(1; 1; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 75 (C\u00e2u 29 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). x+1 y\u22121 z\u22122 Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A (2; 1; 3) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : = \u22122 = . \u0110\u01b0\u1eddng 1 2 th\u1eb3ng \u0111i qua A, vu\u00f4ng g\u00f3c v\u1edbi d v\u00e0 c\u1eaft tr\u1ee5c Oy c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 2t \uf8f1x = 2 + 2t \uf8f1x = 2 + 2t \uf8f1x = 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = \u22123 + 4t. B y=1+t . C y = 1 + 3t . D y = \u22123 + 3t . \uf8f4\uf8f3z = 3t \uf8f4\uf8f3z = 3 + 3t \uf8f4\uf8f3z = 3 + 2t \uf8f3\uf8f4z = 2t \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 \u2206. x+1 y\u22121 z\u22122 G\u0110V\u00ec\u01b0\u1ecdi\u1edd\u2206nMg\u22a5(t0dh;\u1eb3mnn\u00ea;gn0d)A#:\u2208M\u00bbO1\u00b7y#u,\u00bbt==a c0\u00f3\u2212\u21d4A#2M\u2212\u00bb=2=\u2212(2\u22122(2m; m\u2212c\u00f3\u22121V)1T\u2212; C\u22126P3=)l.\u00e00 #u\u00bb = (1; \u22122; 2). \u21d4m= \u22123. \uf8f1x = 2t A# M\u00bb \uf8f4 Do \u0111\u00f3, \u2206 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 = (\u22122; \u22124; \u22123) n\u00ean c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \uf8f2 = \u22123 + 4t . y \uf8f3\uf8f4z = 3t Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 76 (C\u00e2u 35 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). x+1 y z+2 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng \u2206: 2 = \u22121 = 2 v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x+y\u2212 z + 1 = 0. \u0110\u01b0\u1eddng th\u1eb3ng n\u1eb1m trong m\u1eb7t ph\u1eb3ng (P ) \u0111\u1ed3ng th\u1eddi c\u1eaft v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u2206 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 3 + t \uf8f1x = 3 + t \uf8f1x = 3 + 2t \uf8f1x = \u22121 + t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 B y = \u22122 + 4t . C y = \u22122 \u2212 4t. D y = \u22122 + 6t . A y = \u22124t . \uf8f3\uf8f4z = \u22123t \uf8f3\uf8f4z = 2 + t \uf8f3\uf8f4z = 2 \u2212 3t \uf8f3\uf8f4z = 2 + t \u0253 L\u1eddi gi\u1ea3i. \uf8f1x = \u22121 + 2t \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u2206 l\u00e0 y = \u2212t \uf8f4\uf8f3z = \u22122 + 2t. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh \u22121 + 2t \u2212 t \u2212 (\u22122 + 2t) + 1 = 0 ta \u0111\u01b0\u1ee3c t = 2. Suy ra giao \u0111i\u1ec3m c\u1ee7a \u2206 v\u00e0 (P ) l\u00e0 I(3; \u22122; 2). \u2206 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = (2; \u22121; 2) ; (P ) c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb = (1; 1; \u22121). Ta c\u00f3 [ #n\u00bb, #u\u00bb] = (1; \u22124; \u22123). \u0110\u01b0\u1eddng th\u1eb3ng d n\u1eb1m trong (P ) \u0111\u1ed3ng th\u1eddi c\u1eaft v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u2206 s\u1ebd \u0111i qua I \u0111\u1ed3ng th\u1eddi nh\u1eadn [ #n\u00bb, #u\u00bb] \uf8f1x = 3 + t \uf8f4 \uf8f2 l\u00e0m m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng, do \u0111\u00f3 d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 y = \u22122 \u2212 4t \uf8f3\uf8f4z = 2 \u2212 3t. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 551 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian \u0104 C\u00e2u 77 (C\u00e2u 39 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). \uf8f1x = 1 + t \uf8f4 \uf8f2 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : y = 2 + t . G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m \uf8f4\uf8f3z = 3 A(1; 2; 3) v\u00e0 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (0; \u22127; \u22121). \u0110\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c nh\u1ecdn t\u1ea1o b\u1edfi d v\u00e0 \u2206 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = \u22124 + 5t \uf8f1x = \u22124 + 5t \uf8f1x = 1 + 5t \uf8f1x = 1 + 6t \uf8f4\uf8f4\uf8f4 \uf8f4 \uf8f2 \uf8f2\uf8f2\uf8f2 A y = 2 + 11t. B y = \u221210 + 12t. C y = \u221210 + 12t . D y = 2 \u2212 2t . \uf8f4\uf8f3z = 3 + 8t \uf8f4\uf8f3z = 2 + t \uf8f4\uf8f3z = \u22122 + t \uf8f3\uf8f4z = 3 \u2212 t \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d \u0111i qua A(1; 2; 3) v\u00e0 c\u00f3 1 #uv\u00bb\u00e9\u2206c-=t\u01a1(c0h; \u1ec9\u2212p7h; \u01b0\u2212\u01a11n)g\u21d2#u\u00bbd| #u\u00bb=\u2206(|1=; 15;\u221a0)2\u21d2. | #u\u00bbd| = \u221a \u0110\u01b0\u1eddng th\u1eb3ng \u2206 c\u00f3 1 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng 90\u25e6. 2. Ta c\u00f3 #u\u00bbd \u00b7 #u\u00bb\u2206 = \u22127 < 0 \u21d2 ( #u\u00bbd, #u\u00bb\u2206) > \u0110\u01b0\u1eddng ph\u00e2n gi\u00e1c p c\u1ee7a g\u00f3c nh\u1ecdn t\u1ea1o b\u1edfi d \u00c4v\u2212\u00e0 5\u2206\u221ac2\u00f3; \u2212v\u00e91c2-\u221at\u01a12c; h\u2212\u1ec9\u221ap2h\u00e4\u01b0\u01a1hnagy: #\u00bb #a\u00bb = | #u\u00bbd| \u00b7 #u\u00bb\u2206 \u2212 | #u\u00bb\u2206| \u00b7 #u\u00bbd = a = (5; 12; 1). V\u00ec d v\u00e0 \u2206 c\u00f9ng \u0111i qua A(1; 2; 3) n\u00ean \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c p \u0111i qua A. \uf8f1x = \u22124 + 5t \uf8f4 #\u00bb Trong 4 ph\u01b0\u01a1ng \u00e1n, ch\u1ec9 c\u00f3 \u0111\u01b0\u1eddng th\u1eb3ng \uf8f2 = \u221210 + 12t c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng a = (5; 12; 1) v\u00e0 y \uf8f3\uf8f4z = 2 + t \u0111i qua \u0111i\u1ec3m A(1; 2; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 78 (C\u00e2u 35 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng \u2206 : x = y + 1 = z \u2212 1 v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y \u2212 12 1 z + 3 = 0. \u0110\u01b0\u1eddng th\u1eb3ng n\u1eb1m trong (P ) \u0111\u1ed3ng th\u1eddi c\u1eaft v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u2206 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 1 \uf8f1x = \u22123 \uf8f1x = 1 + t \uf8f1x = 1 + 2t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y=1\u2212t . B y = \u2212t . C y = 1 \u2212 2t . D y=1\u2212t . \uf8f4\uf8f3z = 2 + 2t \uf8f4\uf8f3z = 2t \uf8f4\uf8f3z = 2 + 3t \uf8f4\uf8f3z = 2 \u0253 L\u1eddi gi\u1ea3i. \uf8f1x = t \uf8f4 \uf8f2 Ta c\u00f3 \u2206 : y = \u22121 + 2t \uf8f3\uf8f4z = 1 + t. G\u1ecdi M = \u2206 \u2229 (P ) \u21d2 M \u2208 \u2206 \u21d2 M (t; 2t \u2212 1; t + 1). M\u1eb7t kh\u00e1c M \u2208 (P ) \u21d2 t \u2212 2(2t \u2212 1) \u2212 (t + 1) + 3 = 0 \u21d4 t = 1 \u21d2 M (1; 1; 2). V\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) l\u00e0 #n\u00bb = (1; \u22122; \u22121). V\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u2206 l\u00e0 #u\u00bb = (1; 2; 1). \u0110\u01b0\u1eddng th\u1eb3ng d n\u1eb1m trong m\u1eb7t ph\u1eb3ng (P ) \u0111\u1ed3ng th\u1eddi c\u1eaft v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi \u2206 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bbd = [ #n\u00bb, #u\u00bb] = (0; \u22122; 4) hay 1 #u\u00bbd = (0; \u22121; 2). 2 \uf8f1x = 1 \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d : y = 1 \u2212 t \uf8f4\uf8f3z = 2 + 2t. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 552 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 79 (C\u00e2u 38 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). \uf8f1x = 1 + 3t \uf8f4 \uf8f2 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : y = 1 + 4t . G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m \uf8f4\uf8f3z = 1 A (1; 1; 1) v\u00e0 c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng #u\u00bb = (\u22122; 1; 2). \u0110\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c nh\u1ecdn t\u1ea1o b\u1edfi d v\u00e0 \u2206 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = \u221218 + 19t \uf8f1x = \u221218 + 19t \uf8f1x = 1 \u2212 t \uf8f1x = 1 + 27t \uf8f4\uf8f4\uf8f4 \uf8f4 \uf8f2 \uf8f2\uf8f2\uf8f2 A y=1+t . B y = \u22126 + 7t . C y = \u22126 + 7t . D y = 1 + 17t. \uf8f3\uf8f4z = 1 + t \uf8f4\uf8f3z = 11 \u2212 10t \uf8f3\uf8f4z = \u221211 \u2212 10t \uf8f4\uf8f3z = 1 + 10t \u0253 L\u1eddi gi\u1ea3i. \uf8f1x = 1 \u2212 2t \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u2206 : y = 1 + t \uf8f4\uf8f3z = 1 + 2t. D\u1ec5 th\u1ea5y A = d \u2229 \u2206. Ch\u1ecdn B(\u22121; 2; 3) \u2208 \u2206 \u21d2 AB = 3. G\u1ecdi C \u2208 d th\u1ecfa m\u00e3n AB = AC \u21d2 C \u00c5 14 17 \u00e3 ho\u1eb7c C \u00c5 4 ; \u2212 7 ; \u00e3 ; ;1 \u2212 1. 55 55 Ta th\u1ea5y v\u1edbi C \u00c5 4 ; \u2212 7 ; \u00e3 th\u00ec B\u2019AC nh\u1ecdn. \u2212 1 55 Trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n BC \u00c5 9 ; 3 \u00e3 \u0110\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u1ea7n t\u00ecm l\u00e0 IA c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 l\u00e0 I \u2212 ;1 . 10 10 #u\u00bb = (19; 7; \u221210). \uf8f1x = 1 + 19t \uf8f4 \uf8f2 Suy ra \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh y = 1 + 7t d\u1ec5 th\u1ea5y \u0111\u01b0\u1eddng th\u1eb3ng n\u00e0y tr\u00f9ng v\u1edbi \u0111\u01b0\u1eddng \uf8f3\uf8f4z = 1 \u2212 10t \uf8f1x = \u221218 + 19t \uf8f4 \uf8f2 th\u1eb3ng y = \u22126 + 7t \uf8f4\uf8f3z = 11 \u2212 10t. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 80 (C\u00e2u 27 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : x + y + z \u2212 3 = 0 v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : x = y + 1 = 12 z\u22122 \u22121 . H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x+1 y+1 z+1 x\u22121 y\u22121 z\u22121 A \u22121 = \u22124 = . B 3 = \u22122 = \u22121 . 5 x\u22121 y\u22121 z\u22121 x\u22121 y\u22124 z+5 C == \u22125 . D = =. 111 14 \u0253 L\u1eddi gi\u1ea3i. \uf8f1x = t \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 y = \u22121 + 2t \uf8f3\uf8f4z = 2 \u2212 t. \uf8f1x = t \uf8f4 \uf8f4 G\u1ecdi A l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d v\u00e0 m\u1eb7t ph\u1eb3ng (P ). Khi \u0111\u00f3, ta c\u00f3 h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh \uf8f2\uf8f4y = \u22121 + 2t \u21d2 \uf8f4z = 2 \u2212 t \uf8f4 \uf8f4 + y + z \u2212 3 = 0 \uf8f3x t + (\u22121 + 2t) + (2 \u2212 t) \u2212 3 = 0 \u21d4 t = 1 \u21d2 A(1; 1; 1). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 553 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian Ta c\u00f3 \u0111\u01b0\u1eddng th\u1eb3ng d c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bbd = (1; 2; \u22121), m\u1eb7t ph\u1eb3ng (P ) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb(P ) = (1; 1; 1). G\u1ecdi (Q) l\u00e0 m\u1eb7t ph\u1eb3ng ch\u1ee9a \u0111\u01b0\u1eddng th\u1eb3ng d v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ). Khi \u0111\u00f3 (Q) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bb(Q) = #u\u00bbd, #n\u00bb(P ) = (3; \u22122; \u22121). G\u1ecdi \u0111\u01b0\u1eddng th\u1eb3ng \u2206 l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d l\u00ean (P ). Khi \u0111\u00f3 \u2206 l\u00e0 giao tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng (P ) v\u00e0 (Q). Suy ra v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u2206 l\u00e0 #u\u00bb\u2206 = #n\u00bb(P ), #n\u00bb(Q) = (1; 4; \u22125). x\u22121 y\u22121 z\u22121 V\u1eady h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 1 = 4 = \u22125 . Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 81 (C\u00e2u 46 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : x + 1 = y = z \u2212 1 v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x+y\u2212z+3 = 112 0. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ) l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh A x+1 y z\u22121 B x+1 y z\u22121 == . = \u22125 = . 4 5 13 3 1 x\u22121 y z+1 x\u22121 y z+1 C = \u22125 = . D == . 3 1 4 5 13 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi C l\u00e0 giao \u0111i\u1ec3m c\u1ee7a d v\u00e0 (P ). \uf8f1x = \u22121 + t \uf8f4 \uf8f2 Ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a d l\u00e0 y = t \uf8f3\uf8f4z = 1 + 2t. Thay x, y, z v\u00e0o (P ), ta c\u00f3 \u22122 + 2t + t \u2212 1 \u2212 2t + 3 = 0 \u21d2 t = 0. Suy ra to\u1ea1 \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a d v\u00e0 (P ) l\u00e0 C(\u22121; 0; 1). Ta l\u1ea5y A(0; 1; 3) \u2208 d v\u00e0 A = C. G\u1ecdi B l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A tr\u00ean (P ). Suy ra \u0111\u01b0\u1eddng th\u1eb3ng BC l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ). G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) n\u00ean \u2206 c\u00f3 VTCP l\u00e0 #n\u00bb(P) = (2; 1; \u22121). \uf8f1x = 2t \uf8f4 \uf8f2 Suy ra \u2206 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 l\u00e0 y = 1 + t \uf8f4\uf8f3z = 3 \u2212 t. Thay x, y, z v\u00e0o (P ), ta c\u00f3 4t + 1 + t \u2212 3 + t + 3 = 0 \u21d2 t = \u22121. 6 \u00c5 1 5 19 \u00e3 B# C\u00bb \u00c5 2 \u22125; 13 \u00e3 Suy ra giao \u0111i\u1ec3m c\u1ee7a \u2206 v\u00e0 (P ) l\u00e0 B \u2212 ; ; \u21d2 = \u2212 ; \u2212 . 36 6 36 6 B# C\u00bb #a\u00bb = (4; 5; 13). suy ra v\u00e9c-t\u01a1 c\u00f9ng ph\u01b0\u01a1ng v\u1edbi l\u00e0 V\u1eady \u0111\u01b0\u1eddng th\u1eb3ng BC \u0111i qua C nh\u1eadn #a\u00bb = (4; 5; 13) l\u00e0m VTCP c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x+1 y z\u22121 == . 4 5 13 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 82 (C\u00e2u 47 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 1; 3) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : x\u22121 y z+1 == . \u0110\u01b0\u1eddng th\u1eb3ng 121 \u0111i qua A, c\u1eaft tr\u1ee5c Oy v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 1 + t \uf8f1x = \u22123 + 3t \uf8f1x = 1 + t \uf8f1x = \u22121 + t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = 1 + 2t. B y = 4 \u2212 2t . C y = 1 \u2212 t . D y = 5 \u2212 2t . \uf8f4\uf8f3z = 3 + 3t \uf8f4\uf8f3z = \u22121 + t \uf8f3\uf8f4z = 3 + t \uf8f3\uf8f4z = \u22123 + 3t \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 554 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng ct\u00f3h\u1ecfA#aB\u00bbm=\u00e3n(\u2212y\u00ea1u; c\u1ea7u \u0111\u1ec1 b\u00e0i, B(0; b; 0) l\u00e0 giao \u0111i\u1ec3m \u2206 c\u1ee7a \u2206 v\u00e0 tr\u1ee5c Oy. Ta b \u2212 1; \u22123). y #u\u00bb = d TS\u0110uh\u01b0ye\u1eddonrag\u0111\u1ec1At#hbB\u1eb3\u00bb\u00e0ni\u00b7gt#u\u00bbad=cc\u00f3\u00f30A#m\u21d4B\u1ed9\u00bbt\u2212\u22a5v1\u00e9#uc\u00bb+-.t2\u01a1(cbh\u2212\u1ec9 ph\u01b0\u01a1ng l\u00e0 \u21d4b (1; 2; 1). A 1) \u2212 3 = 0 = 3. O B Suy ra A# B\u00bb = (\u22121; 2; \u22123) \u21d2 #v\u00bb = (1; \u22122; 3) l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u2206. \uf8f1x = 1 + t \uf8f4 \uf8f2 Suy ra ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u2206 l\u00e0 y = 1 \u2212 2t \uf8f3\uf8f4z = 3 + 3t. Khi t = \u22122 th\u00ec \u2206 \u0111i qua \u0111i\u1ec3m M (\u22121; 5; \u22123). \uf8f1x = \u22121 + t \uf8f4 \uf8f2 Hay \u2206 c\u0169ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 y = 5 \u2212 2t \uf8f3\uf8f4z = \u22123 + 3t. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 83 (C\u00e2u 46 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A (3; 1; 1) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : x\u22121 y z+1 == . \u0110\u01b0\u1eddng th\u1eb3ng 121 \u0111i qua A, c\u1eaft tr\u1ee5c Oy v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 3 + t \uf8f1x = \u22121 + t \uf8f1x = 3 + 3t \uf8f1x = \u22123 + 3t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = 1\u2212t. B y = 4 \u2212 2t . C y = 1 \u2212 t . D y = 5 \u2212 2t . \uf8f3\uf8f4z = 1 + t \uf8f3\uf8f4z = \u22123 + 3t \uf8f4\uf8f3z = 1 + t \uf8f3\uf8f4z = \u22121 + t \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm. G\u1ecdi B = \u2206 \u2229 Oy \u21d2 \u00aeA# B\u00bb = (\u22123; b \u2212 1; \u22121) B (0; b; 0) \u21d2 #u\u00bbd = SXTua\u00e9ytc\u00f3\u0111rai\u2206\u1ec3Am# \u22a5B\u00bbMd=(\u21d2\u2212(\u22123A#3; B5;\u00bb2; \u2212;\u22a5\u221211)#u\u00bb,)dt\u21d2a\u21d4c#u\u00f3\u00bbA#\u2206BA#\u00bb=M\u00b7\u00bb#u\u00bb(=3d;=\u2212(\u2212206; \u21d41; 4).;(\u2212\u221223)) \u00b71 + (b \u2212 1) \u00b7 2 + (\u22121) \u00b7 (1; 2; 1) . 0 \u21d4 b = 3. v\u00e0 1 = 0 \u21d4 2b \u2212 6 = M\u00bb, hay ba B# M\u00bb = (\u22123; 2; \u22121). Suy ra A# M\u00bb = 2B# \u0111i\u1ec3m A, B, M th\u1eb3ng h\u00e0ng \u21d2 M \u2208 \u2206. \u0110\u01b0\u1eddng th\u1eb3ng \u2206 \u0111i qua \u0111i\u1ec3m M , nh\u1eadn #u\u00bb\u2206 = (3; \u22122; 1) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = \u22123 + 3t \uf8f4 \uf8f2 y = 5 \u2212 2t \uf8f3\uf8f4z = \u22121 + t. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 84 (C\u00e2u 43 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). x\u22121 y z+1 Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 1; 1) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : == . \u0110\u01b0\u1eddng th\u1eb3ng 121 qua A, c\u1eaft tr\u1ee5c Oy v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 1 \u2212 3t \uf8f1x = \u22121 + t \uf8f1x = \u22121 + t \uf8f1x = 1 + t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y=1+t . B y=2+t . C y=3\u2212t . D y = 1 \u2212 2t . \uf8f4\uf8f3z = 1 + t \uf8f4\uf8f3z = 3 \u2212 3t \uf8f4\uf8f3z = \u22121 + t \uf8f4\uf8f3z = 1 + t Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 555","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian DGK\u0110Do\u01b0ohi\u1ea3\u1eddi\u0111\u0111ns\u0111\u01b0\u00f3\u1eedg\u00f3\u1edd\u0111\u0111ntA#\u01b0h\u01b0gB\u1edd\u1eb3\u1edd\u00bbtnnnhg=gg\u1eb3tdnt(hhg\u2212c\u1eb3\u1eb3\u00f31cnn\u1ea7;gmg1nc;\u1ed9ct\u1ea7\u2212\u1ea7t\u00ecnmn1v)t\u00e9t.\u00ecvc\u00ecmm-ut\u00f4\u01a1ccn\u00f3\u1eafcghtmg\u1ec9t\u00f3\u1ed9rptc\u1ee5hcv\u01b0v\u00e9\u1edb\u01a1Ocin-ygdt\u01a1tln\u1ea1\u00e0c\u00eaihn#u\u0111\u00bb\u1ec9 iA#p\u1ec3=mhB\u00bb\u01b0(1\u01a1\u00b7B;n#u\u00bb(2g0;=;l1\u00e0b);0.A0#\u21d4)B,\u00bbt\u2212=a 1c(\u00f31+;A#\u22122B(1\u00bbb;\u2212=1).1(\u2212) \u22121; b \u2212 1; \u22121). 2. 1 = 0\u21d4b= \uf8f1x = 1 + t \uf8f4 \uf8f2 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A, c\u1eaft tr\u1ee5c Oy v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d l\u00e0 y = 1 \u2212 t \uf8f3\uf8f4z = 1 + t. D\u1ec5 th\u1ea5y \u0111\u01b0\u1eddng th\u1eb3ng tr\u00ean \u0111i qua \u0111i\u1ec3m C(\u22121; 3; \u22121) n\u00ean ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A, c\u1eaft tr\u1ee5c \uf8f1x = \u22121 + t \uf8f4 \uf8f2 Oy v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d l\u00e0 y = 3 \u2212 t \uf8f4\uf8f3z = \u22121 + t. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 85 (C\u00e2u 48 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 3; 1) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : x\u22121 y z+1 == . \u0110\u01b0\u1eddng th\u1eb3ng 121 \u0111i qua A, c\u1eaft tr\u1ee5c Oy v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = 1 + t \uf8f1x = \u22121 \u2212 t \uf8f1x = 2 \u2212 t \uf8f1x = 1 + 3t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = 3+t. B y=1\u2212t . C y = 2+t. D y=3\u2212t . \uf8f4\uf8f3z = 1 + t \uf8f3\uf8f4z = 3 + 3t \uf8f4\uf8f3z = 2 \u2212 t \uf8f4\uf8f3z = 1 \u2212 t \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = (1; 2; 1). G\u1ecdi \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 \u2206 v\u00e0 B = \u2206 \u2229 Oy. Ta c\u00f3 B \u2208 Oy suy ra B(0; b; 0) (v\u1edbi b \u2208 R). V\u00ec A, B \u2208 \u2206 n\u00ean \u2206 \u22a5 d \u21d4 A# B\u00bb \u22a5 #u\u00bb \u21d4 A# B\u00bb \u00b7 #u\u00bb = 0. (1) Ta c\u00f3 #\u00bb = (\u22121; b \u2212 3; \u22121). AB T\u1eeb \u0111\u00f3 (A#1)B\u00bb\u21d4=\u2212(\u22121 1+; 12;(\u2212b \u22121).3) \u2212 1 = 0 \u21d4 b = 4. Suy ra \uf8f1x = 1 \u2212 t Khi \u0111\u00f3 \u0111\u01b0\u1eddng th\u1eb3ng \u2206 qua A v\u00e0 nh\u1eadn A# B\u00bb l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng n\u00ean c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u2206 : \uf8f4 \uf8f2 y = 3+t \uf8f3\uf8f4z = 1 \u2212 t. \uf8f1x = 2 \u2212 t \uf8f4 \uf8f2 Khi t = \u22121 ta \u0111\u01b0\u1ee3c C(2; 2; 2) \u2208 \u2206 n\u00ean \u2206 c\u0169ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u2206 : y = 2 + t \uf8f4\uf8f3z = 2 \u2212 t. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 86 (C\u00e2u 46 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(\u22124; \u22123; 3) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x + y + z = 0. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A, c\u1eaft tr\u1ee5c Oz v\u00e0 song song v\u1edbi (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 A x\u22124 y\u22123 z\u22123 B x+4 y+3 z\u22123 4 = 3 = \u22127 . = =. 431 x+4 y+3 z\u22123 x + 8 y + 6 z \u2212 10 C \u22124 = = . D 4 = 3 = \u22127 . 3 1 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 556","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN G\u1ecdi d l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng th\u1ecfa \u0111\u1ec1 b\u00e0i. \u0110\u1eb7t M (0; 0; m) = d \u2229 Oz. A M #n\u00bb = P d - M\u1eb7t pl\u00e0h\u1eb3#u\u00bbng=(PA# )M\u00bbc\u00f3=V(4T;P3;Tml\u00e0\u2212 3). (1; 1; 1), \u0111\u01b0\u1eddng th\u1eb3ng d c\u00f3 VTCP - V\u00ec d \u2225 (P ) \u21d2 #u\u00bb \u22a5 #n\u00bb \u21d4 #u\u00bb \u00b7 #n\u00bb = 0 \u21d4 4 + 3 + m \u2212 3 = 0 \u21d4 #n\u00bb m = \u22124. - d c\u00f3 VTCP l\u00e0 #u\u00bb = (4; 3; \u22127) n\u00ean lo\u1ea1i \u0111\u01b0\u1ee3c c\u00e1c ph\u01b0\u01a1ng \u00e1n x+4 y+3 z\u22123 x+4 y+3 z\u22123 = = v\u00e0 \u22124 = = . - 4 3 1 A(\u22124; \u22123; 3) 31 #u\u00bb = (4; 3; \u22127) \u0110\u01b0\u1eddng th\u1eb3ng d qua v\u00e0 c\u00f3 VTCP x+4 y+3 z\u22123 n\u00ean d c\u00f3 PTCT l\u00e0: 4 = 3 = \u22127 . - V\u00ec d \u0111i qua \u0111i\u1ec3m N (\u22128; \u22126; 10) n\u00ean x + 8 = y + 6 = z \u2212 10 l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a d. 4 3 \u22127 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 87 (C\u00e2u 29 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho hai \u0111\u01b0\u1eddng th\u1eb3ng x\u22123 y\u22123 z+2 x\u22125 y+1 z\u22122 d1 : \u22121 = \u22122 = 1 ; d2 : \u22123 = 2 = 1 v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x + 2y + 3z \u2212 5 = 0. \u0110\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi (P ), c\u1eaft d1 v\u00e0 d2 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x\u22121 y+1 z x\u22122 y\u22123 z\u22121 A = = . B = =. 1 23 123 x\u22123 y\u22123 z+2 x\u22121 y+1 z C == . D = =. 123 3 21 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n vi\u1ebft l\u00e0 \u2206, A l\u00e0 giao c\u1ee7a \u2206 v\u00e0 d1, B l\u00e0 giao c\u1ee7a \u2206 v\u00e0 d2. Khi \u0111\u00f3 A(3 \u2212 t; 3 \u2212 2t; \u22122 + t) v\u00e0 B(5 \u2212 3u; \u22121 + 2u; 2 + u). Suy ra A# B\u00bb = (2 + t \u2212 3u; \u22124 + 2u + 2t; 4 + u \u2212 t). M\u1eb7t ph\u1eb3ng (P ) c\u00f3 m\u1ed9t VTPT l\u00e0 #n\u00bb = (1; 2; 3). Ta c\u00f3 \u2206 \u22a5 (P ) \u21d2 #\u00bb v\u00e0 #n\u00bb c\u00f9ng ph\u01b0\u01a1ng. Do \u0111\u00f3: AB 2 + t \u2212 3u = \u22124 + 2u + 2t = 4 + u \u2212 t \u21d2 \u00aet = 2 1 2 3 u = 1. Suy ra A(1; \u22121; 0), ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u2206: x\u22121 = y+1 = z . 1 23 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 88 (C\u00e2u 49 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) song song v\u00e0 c\u00e1ch \u0111\u1ec1u x\u22122 y z x y\u22121 z\u22122 hai \u0111\u01b0\u1eddng th\u1eb3ng d1 : \u22121 = 1 = 1 v\u00e0 d2 : 2 = \u22121 = \u22121 . A (P ) : 2x \u2212 2z + 1 = 0. B (P ) : 2y \u2212 2z + 1 = 0. C (P ) : 2x \u2212 2y + 1 = 0. D (P ) : 2y \u2212 2z \u2212 1 = 0. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 d1 \u0111i qua \u0111i\u1ec3m A(2; 0; 0) v\u00e0 c\u00f3 VTCP #u\u00bb1 = (\u22121; 1; 1). d2 \u0111i qua \u0111i\u1ec3m B(0; 1; 2) v\u00e0 c\u00f3 VTCP #u\u00bb2 = (2; \u22121; \u22121). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 557 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian V\u00ec (P ) song song v\u1edbi hai \u0111\u01b0\u1eddng th\u1eb3ng d1 v\u00e0 d2 n\u00ean VTPT c\u1ee7a (P ) l\u00e0 #n\u00bb = [ #u\u00bb1, #u\u00bb2] = (0; 1; \u22121). Khi \u0111\u00f3 (P ) c\u00f3 d\u1ea1ng y \u2212 z + D = 0 \u21d2 lo\u1ea1i \u0111\u00e1p \u00e1n A v\u00e0 C. \u00c51\u00e3 c\u1ee7a AB. L\u1ea1i c\u00f3 (P ) c\u00e1ch \u0111\u1ec1u d1 v\u00e0 d2 n\u00ean (P ) \u0111i qua trung \u0111i\u1ec3m M 0; ; 1 2 Do \u0111\u00f3 P : 2y \u2212 2z + 1 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 89 (C\u00e2u 33 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho c\u00e1c \u0111i\u1ec3m A(2; \u22121; 0), B(1; 2; 1), C(3; \u22122; 0) v\u00e0 D(1; 1; \u22123). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua D v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \uf8f1x = t \uf8f1x = t \uf8f1x = 1 + t \uf8f1x = 1 + t . \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y=t . B y=t . C y=1+t . D y=1+t \uf8f4\uf8f3z = \u22121 \u2212 2t \uf8f3\uf8f4z = 1 \u2212 2t \uf8f3\uf8f4z = \u22122 \u2212 3t \uf8f4\uf8f3z = \u22123 + 2t \u0253 L\u1eddi gi\u1ea3i. #\u00bb #\u00bb \u00ee# \u00bb # \u00bb\u00f3 Ta c\u00f3 AB = (\u22121; 3; 1); AC = (1; \u22121; 0); #n\u00bb(ABC) = AB, AC = (1; 1; \u22122). \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua D v\u00e0 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: \uf8f1x = 1 + t \uf8f4 #n\u00bb(ABC) = (1; 1; \u22122) v\u1eady ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 l\u00e0: \uf8f2 = 1 + t y \uf8f3\uf8f4z = \u22123 \u2212 2t. \uf8f1x = t \uf8f4 \uf8f2 \u0110\u01b0\u1eddng th\u1eb3ng n\u00e0y c\u0169ng ch\u00ednh l\u00e0 y = t \uf8f3\uf8f4z = \u22121 \u2212 2t. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 90 (C\u00e2u 45 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). x y\u22121 z\u22122 Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : = = \u22121 v\u00e0 m\u1eb7t ph\u1eb3ng 1 1 (P ) : x + 2y + z \u2212 4 = 0. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ) l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh x y+1 z+2 x y+1 z+2 A = 1 = \u22124 . B = \u22122 = . 2 3 1 x y\u22121 z\u22122 x y\u22121 z\u22122 C = = \u22124 . D = = . 2 1 3 \u22122 1 \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (P ) c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 #n\u00bbP = (1; 2; 1). \uf8f1x = t \uf8f4 \uf8f2 G\u1ecdi M l\u00e0 giao \u0111i\u1ec3m c\u1ee7a d : y = 1 + t v\u00e0 (P ). \uf8f3\uf8f4z = 2 \u2212 t Do M \u2208 d \u21d2 M (m; m + 1; \u2212m + 2). M\u1eb7t kh\u00e1c M \u2208 (P ) \u21d4 m + 2(m + 1) + (\u2212m + 2) \u2212 4 = 0 \u21d4 m = 0. Suy ra M (0; 1; 2). L\u1ea5y N (1; 2; 1) \u2208 d, g\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng qua N v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ). Suy ra \u0111\u01b0\u1eddng th\u1eb3ng \u2206 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb\u2206 = #n\u00bbP = (1; 2; 1). \uf8f1x = 1 + t \uf8f4 \uf8f2 Do \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u2206 l\u00e0 y = 2 + 2t \uf8f4\uf8f3z = 1 + t. G\u1ecdi H l\u00e0 giao \u0111i\u1ec3m \u2206 v\u00e0 (P ). Do H \u2208 d \u21d2 M (1 + h; 2 + 2h; 1 + h). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 558 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN M\u1eb7t kh\u00e1c H \u2208 (P ) \u21d4 1 + h + 2(2 + 2h) + (1 + h) \u2212 4 = 0 \u21d4 6h + 2 = 0 \u21d4 h = \u22121 . 3 \u00c52 4 2\u00e3 Suy ra H ; ; . 333 Ta c\u00f3 M# H\u00bb = \u00c52 1 \u2212 4 \u00e3 ; ; . 33 3 G\u1ecdi d h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ). v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 y#u\u00bb\u2212d = 3M# H\u00bb = (2; 1; \u22124). Suy ra \u0111\u01b0\u1eddng th\u1eb3ng d qua M (0; 1; 2) c\u00f3 m\u1ed9t x 1 z\u22122 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c d c\u1ee7a d tr\u00ean (P ) l\u00e0: = 1 = \u22124 . 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 91 (C\u00e2u 43 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d: x\u22121 = y\u22122 = z+1 v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x + 2y \u2212 1 1 \u22122 z \u2212 6 = 0. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ) l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh A x+1 y+2 z\u22121 B x\u22121 y\u22122 z+1 = \u22121 = . = \u22121 = . 3 1 3 1 x+1 y+2 z\u22121 x\u22121 y\u22122 z+1 C \u22121 = 4 =. D \u22121 = = . 4 7 7 \u0253 L\u1eddi gi\u1ea3i. X\u00e9t A(1; 2; \u22121) \u2208 d. D\u1ec5 th\u1ea5y A \u2208 (P ). \uf8f1x = 2 + t \uf8f4 \uf8f2 X\u00e9t B(2; 3; \u22123) \u2208 d. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u2206 \u0111i qua B v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) : y = 3 + 2t \uf8f3\uf8f4z = \u22123 \u2212 t. Thay v\u00e0o ph\u01b0\u01a1ng tr\u00ecnh (P ) : 2 + t + 6 + 4t + 3 + t \u2212 6 = 0 \u21d4 t = \u22125 Suy ra giao \u0111i\u1ec3m c\u1ee7a \u2206 v\u00e0 (P ), . 6 \u00c57 8 \u221213\u00e3 ch\u00ednh l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a B tr\u00ean (P ), l\u00e0 H ; ; . 66 6 Ta c\u00f3 A# H\u00bb = \u00c51 \u22124 \u22127 \u00e3 Do \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ) l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng ; ; . 66 6 x\u22121 y\u22122 z+1 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u22121 = = . 4 7 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 92 (C\u00e2u 45 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(0; 4; \u22123). X\u00e9t \u0111\u01b0\u1eddng th\u1eb3ng d thay \u0111\u1ed5i, song song v\u1edbi tr\u1ee5c Oz v\u00e0 c\u00e1ch tr\u1ee5c Oz m\u1ed9t kho\u1ea3ng b\u1eb1ng 3. Khi kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn d l\u1edbn nh\u1ea5t, d \u0111i qua \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y ? A P (\u22123; 0; \u22123). B Q(0; 11; \u22123). C N (0; 3; \u22125). D M (0; \u22123; \u22125). \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 559 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian z O d 4y x A HA V\u00ec d thay \u0111\u1ed5i, song song v\u1edbi tr\u1ee5c Oz v\u00e0 c\u00e1ch tr\u1ee5c Oz m\u1ed9t kho\u1ea3ng b\u1eb1ng 3 n\u00ean d l\u00e0 \u0111\u01b0\u1eddng sinh c\u1ee7a h\u00ecnh tr\u1ee5 c\u00f3 tr\u1ee5c l\u00e0 Oz v\u00e0 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 3. G\u1ecdi A l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a A l\u00ean tr\u1ee5c Oz, d\u1ec5 th\u1ea5y A (0; 0; \u22123) v\u00e0 AA = 4. G\u1ecdi H(x; y; z) l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a A l\u00ean d. AH l\u1edbn nh\u1ea5t khi A, A , H th\u1eb3ng h\u00e0ng v\u00e0 AH = AA + A H = AA + r = 4 + 3 = 7. \uf8f1x = 0 A# H\u00bb 7# \u00bb 7(0; \u22124; 0) \uf8f4 Khi \u0111\u00f3 = AA \u21d4 (x; y \u2212 4; z + 3) = \u21d4 \uf8f2 = \u22123 \u21d2 H(0; \u22123; \u22123). y 4 4 \uf8f4\uf8f3z = \u22123 \uf8f1x = 0 #\u00bb \uf8f4 V\u1eady d qua H(0; \u22123; \u22123) c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng k = (0; 0; 1) n\u00ean c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \uf8f2 = \u22123 suy ra y \uf8f4\uf8f3z = \u22123 + t d \u0111i qua \u0111i\u1ec3m M (0; \u22123; \u22125). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 93 (C\u00e2u 20 - 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 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m A(2; 3; 0) v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (P ) : x + 3y \u2212 z + 5 = 0? \uf8f1x = 1 + 3t \uf8f1x = 1 + t \uf8f1x = 1 + t \uf8f1x = 1 + 3t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = 3t B y = 3t C y = 1 + 3t D y = 3t \uf8f4\uf8f3z = 1 \u2212 t. \uf8f4\uf8f3z = 1 \u2212 t. \uf8f3\uf8f4z = 1 \u2212 t. \uf8f3\uf8f4z = 1 + t. \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (P ) nh\u1eadn n# (P\u00bb) = (1; 3; \u22121) l\u00e0m v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng \u21d2 ph\u01b0\u01a1ng \uf8f1x = 2 + t \uf8f4 \uf8f2 tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 y = 3 + 3t . \uf8f3\uf8f4z = \u2212t \uf8f1x = 1 + t \uf8f4 \uf8f2 L\u1ea5y t = \u22121 \u21d2 N (1; 0; 1) thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng \u21d2 \u0111\u00e1p \u00e1n \u0111\u00fang l\u00e0: y = 3t . \uf8f3\uf8f4z = 1 \u2212 t Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 94 (C\u00e2u 34 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m M (\u22121; 1; 3) v\u00e0 hai \u0111\u01b0\u1eddng th\u1eb3ng \u2206 : x \u2212 1 = 3 y+3 z\u22121 x+1 y z = ,\u2206 : = 3 = \u22122 . Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng 2 1 1 \u0111i qua M , vu\u00f4ng g\u00f3c v\u1edbi \u2206 v\u00e0 \u2206 ? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 560 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \uf8f1x = \u22121 \u2212 t \uf8f1x = \u2212t \uf8f1x = \u22121 \u2212 t \uf8f1x = \u22121 \u2212 t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y=1+t B y=1+t C y=1\u2212t D y=1+t \uf8f4\uf8f3z = 1 + 3t. \uf8f3\uf8f4z = 3 + t. \uf8f4\uf8f3z = 3 + t. \uf8f3\uf8f4z = 3 + t. \u0253 L\u1eddi gi\u1ea3i. \u2206 v\u00e0 \u2206 c\u00f3 c\u00e1c v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u1ea7n l\u01b0\u1ee3t l\u00e0 u#\u00bb1 = (3; 2; 1) v\u00e0 u#\u00bb2 = (1; 3; \u22122). ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 Khi \u0111\u00f3 [u#\u00bb1, u#\u00bb2] = (\u22127; 7; 7) \u21d2 \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi d v\u00e0 \u2206 c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 \uf8f1x = \u22121 \u2212 t \uf8f4 #u\u00bb = (\u22121; 1; 1) \u21d2 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \uf8f2 = 1 + t . y \uf8f3\uf8f4z = 3 + t Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 95 (C\u00e2u 37 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). x\u22121 \uf8f1x = 1 + 3t d2 : 2 = \uf8f4 \uf8f2 Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111\u01b0\u1eddng th\u1eb3ng d1 : y = \u22122 + t, \uf8f3\uf8f4z = 2 y+2 = z v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x + 2y \u2212 3z = 0. Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t \u22121 2 ph\u1eb3ng \u0111i qua giao \u0111i\u1ec3m c\u1ee7a d1 v\u00e0 (P ), \u0111\u1ed3ng th\u1eddi vu\u00f4ng g\u00f3c v\u1edbi d2? A 2x \u2212 y + 2z + 22 = 0. B 2x \u2212 y + 2z + 13 = 0. C 2x \u2212 y + 2z \u2212 13 = 0. D 2x + y + 2z \u2212 22 = 0. \u0253 L\u1eddi gi\u1ea3i. Giao c\u1ee7a d1 v\u00e0 (P ) l\u00e0 \u0111i\u1ec3m M (4; \u22121; 2). C\u00e1c m\u1eb7t ph\u1eb3ng trong 4 ph\u01b0\u01a1ng \u00e1n c\u00f9ng vu\u00f4ng g\u00f3c v\u1edbi d2 nh\u01b0ng ch\u1ec9 c\u00f3 m\u1eb7t ph\u1eb3ng \u1edf ph\u01b0\u01a1ng \u00e1n C \u0111i qua M (4; \u22121; 2) n\u00ean ch\u1ecdn C. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 96 (C\u00e2u 19 - 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 (1; \u22122; \u22123), B (\u22121; 4; 1) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : x+2 y\u22122 z+3 = \u22121 = . Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng 1 2 \u0111i qua trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB v\u00e0 song song v\u1edbi d? y\u22121 x y\u22122 z+2 A x = z+1 B = = . = . 1 \u22121 2 11 2 x y\u22121 z+1 x\u22121 y\u22121 z+1 C = \u22121 = . D = \u22121 = . 1 2 1 2 \u0253 L\u1eddi gi\u1ea3i. Trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n AB l\u00e0 M (0; 1; \u22121), x\u00e9t d c\u00f3 v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 #u\u00bb = (1; \u22121; 2) \u21d2 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng qua M v\u00e0 song song v\u1edbi d l\u00e0 x = y\u22121 = z +1 1 \u22121 . 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 97 (C\u00e2u 33 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111i\u1ec3m A(1; \u22121; 2), B(\u22121; 2; 3) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d : x\u22121 = y\u22122 = z\u22121 T\u00ecm \u0111i\u1ec3m M (a; b; c) thu\u1ed9c d sao cho M A2 + M B2 = 28, bi\u1ebft . 112 c < 0. A M (\u22121; 0; \u22123). B M (2; 3; 3). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 561 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian \u00c5 1 7 2 \u00e3 \u00c5 1 7 2 \u00e3 . \u2212 . C M ; ; \u2212 D M ; \u2212 ; \u2212 66 3 663 \u0253 L\u1eddi gi\u1ea3i. V\u00ec M \u2208 d n\u00ean t\u1ecda \u0111\u1ed9 M c\u00f3 d\u1ea1ng M (1 + t; 2 + t; 1 + 2t). Ta c\u00f3 M A2 + M B2 = 28 \u21d4 t2 + (t + 3)2 + (2t \u2212 1)2 + (t + 2)2 + t2 + (2t \u2212 2)2 = 28 \u21d4 12t2 \u2212 2t \u2212 10 = 0 \u21d4 t = 1; t = \u2212 5 . 6 V\u1edbi t = 1 \u21d2 M (2; 3; 3) lo\u1ea1i v\u00ec c < 0. V\u1edbi t = \u22125 \u21d2 M \u00c51 7;\u22122\u00e3 th\u1ecfa y\u00eau c\u1ea7u b\u00e0i to\u00e1n. ; 6 66 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 98 (C\u00e2u 50 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) t\u00e2m I(1; 4; 2), b\u00e1n k\u00ednh b\u1eb1ng 2. G\u1ecdi M, N l\u00e0 hai \u0111i\u1ec3m l\u1ea7n l\u01b0\u1ee3t thu\u1ed9c hai tr\u1ee5c Ox, Oy sao cho \u0111\u01b0\u1eddng th\u1eb3ng M N ti\u1ebfp x\u00fac v\u1edbi (S), \u0111\u1ed3ng th\u1eddi m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OIM N c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 7 \u00b7 G\u1ecdi A l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a M N v\u00e0 (S), gi\u00e1 tr\u1ecb AM.AN 2 b\u1eb1ng \u221a B 14. \u221a D 8. A 9 2. C 6 2. \u0253 L\u1eddi gi\u1ea3i. z I B O Mx A N y G\u1ecdi M (a; 0; 0) \u2208 Ox, N (0; b; 0) \u2208 Oy. Ta c\u00f3 d I; (Oxy) = 2 = R n\u00ean (S) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy) t\u1ea1i \u0111i\u1ec3m A(1; 4; 0) v\u00e0 M N c\u0169ng \u0111i qua A. # \u00bb = (a \u2212 1; \u22124; 0), #\u00bb = (\u22121; b \u2212 4; 0) v\u00e0 3 \u0111i\u1ec3m A, M, N th\u1eb3ng h\u00e0ng n\u00ean ta \u0111\u01b0\u1ee3c L\u1ea1i c\u00f3 AM AN a\u22121 = \u22124 \u21d4 (a \u2212 1)(b \u2212 4) = 4. (1) \u22121 b\u22124 T\u1ee9 di\u1ec7n OIM N c\u00f3 IA \u22a5 (OM N ) v\u00e0 OM N vu\u00f4ng t\u1ea1i O n\u00ean n\u1ebfu g\u1ecdi J l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OIM N th\u00ec J \u2208 (IM N ). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 562 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN Suy ra b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OIM N b\u1eb1ng b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp IMN. (2) Ta c\u00f3 S IMN = IM.IN.M N (v\u1edbi r = 7 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp IM N ). 2 4r \u21d41 IA.M N = IM.IN.M N 2 4\u00b7 7 2 \u21d4IM.IN = 7IA \u21d4 IM.IN = 14 \u21d4 (a \u2212 1)2 + 20 . (b \u2212 4)2 + 5 = 196. \u00df m=a\u22121 n = b \u2212 4. \u0110\u1eb7t \uf8f14 \uf8f4\uf8f4n = (3) \u00aem.n = 4 \uf8f2 m (4) T\u1eeb (1) v\u00e0 (2) ta c\u00f3 h\u1ec7 m2 + 20 n2 + 5 = 196 \u21d4 \u00c5 16 \u00e3 m2 + 5 = 196. \uf8f4 m2 + 20 \uf8f4 \uf8f3 T\u1eeb (4) ta \u0111\u01b0\u1ee3c m2 + 20 . 16 + 5m2 = 196m2 \u00ef \u221a\u221a m = 2 \u221a2 \u00ef n = \u221a2 m = \u22122 2 n = \u2212 2. \u21d4 5m4 \u2212 80m2 + 320 = 0 \u21d4 m2 = 8 \u21d4 \u21d2 \u00ef \u221a\u221a a = 1 + 2\u221a2, b = 4 + \u221a2 Suy ra a = 1 \u2212\u221a2 2, b = 4 \u2212 2. V\u1eady AM.AN = 6 2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 99 (C\u00e2u 43 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : x = y = z\u22121 v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : x + 2y \u2212 2z + 2 = 1 \u22121 2 0. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a d tr\u00ean (P ) l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh A x y z\u22121 B x y z+1 C x y z+1 D x y z\u22121 \u22122 = 4 = . == . \u22122 = 4 = . == . 3 14 1 8 3 14 1 8 \u0253 L\u1eddi gi\u1ea3i. Gi\u1ea3 s\u1eed \u2206 l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a d tr\u00ean (P ). #n\u00bbP d P G\u1ecdi (Q) l\u00e0 m\u1eb7t ph\u1eb3ng ch\u1ee9a \u2206 v\u00e0 d. X\u00e9t I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a d v\u00e0 (P ) th\u00ec I \u2208 \u2206 to\u1ea1 \u0111\u1ed9 c\u1ee7a I #u\u00bbd Q I l\u00e0 nghi\u1ec7m c\u1ee7a h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh #n\u00bbQ #u\u00bb\u2206 \u2206 \uf8f1x y z \u2212 1 MDD-109 \uf8f2 = \u22121 = 2 1 \uf8f3x + 2y \u2212 2z + 2 = 0 \u21d2 x = 2y = 2z \u2212 2 = x + 2y \u2212 2z + 2 = 0 = 0 \u21d2x= 0, y = 0, z = 1\u21d2 I(0; 0; 1). (1) 1 \u22122 4 1\u22122\u22124 \u22125 X\u00e9t #u\u00bbd = (1; \u22121; 2), #n\u00bbP = (1; 2; \u22122), ta c\u00f3 \u00ae(Q) \u2283 d \u21d2 \u00ae #n\u00bbQ \u22a5 #u\u00bbd . Ch\u1ecdn #n\u00bbQ = [ #u\u00bbd, #n\u00bbP ] = (\u22122; 4; 3). (Q) \u22a5 (P ) #n\u00bbQ \u22a5 #n\u00bbP M\u1eb7t kh\u00e1c \u00ae #u\u00bb\u2206 #n\u00bbP #u\u00bb\u2206 #n\u00bbQ \u00ae\u2206 \u2282 (P ) \u21d2 \u22a5 . Ch\u1ecdn #u\u00bb\u2206 = [ #n\u00bbP , #n\u00bbQ] = (14; 1; 8). (2) \u2206 \u2282 (Q) \u22a5 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 563 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian x y z\u22121 T\u1eeb (1) v\u00e0 (2), ta \u0111\u01b0\u1ee3c \u2206 : = = . 14 1 8 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 100 (C\u00e2u 49 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 2)2 + (y \u2212 3)2 + (z + 1)2 = 16 v\u00e0 \u0111i\u1ec3m A (\u22121; \u22121; \u22121). 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 3x + 4y \u2212 2 = 0. B 3x + 4y + 2 = 0. C 6x + 8y + 11 = 0. D 6x + 8y \u2212 11 = 0. \u0253 L\u1eddi gi\u1ea3i. MC\u00f3\u1eb7tI#Ac\u00bb\u1ea7u= (S) c\u00f3 t\u00e2m I(2; 3; \u22121) b\u00e1n k\u00ednh R = 4. M (\u22123; \u22124; 0) \u21d2 IA = 5. M\u1eb7t ph\u1eb3ng c\u1ed1 \u0111\u1ecbnh#\u0111\u00bbi qua \u0111i\u1ec3m H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a M xu\u1ed1ng IA v\u00e0 nh\u1eadn IA l\u00e0m v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn Do hai M HI v\u00e0 AM I \u0111\u1ed3ng d\u1ea1ng n\u00ean I A H IM2 = IH \u00b7 IA \u21d2 IH = IM2 = 16 . IA 5 I#H\u00bb 16 I# A\u00bb \u00c5 2 11 \u00e3 \u22121 . Suy ra = \u21d2 H ; ; 25 25 25 N \u00c5 2 \u00e3 \u00c5 11 \u00e3 M\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u22123 x \u2212 y \u2212 4 \u2212 = 0 \u21d4 3x + 4y \u2212 2 = 0. 25 25 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 101 (C\u00e2u 42 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(0; 4; \u22123). X\u00e9t \u0111\u01b0\u1eddng th\u1eb3ng d thay \u0111\u1ed5i, song song v\u1edbi tr\u1ee5c Oz v\u00e0 c\u00e1ch tr\u1ee5c Oz m\u1ed9t kho\u1ea3ng b\u1eb1ng 3. Khi kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn d nh\u1ecf nh\u1ea5t, d \u0111i qua \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A P (\u22123; 0; \u22123). B M (0; \u22123; \u22125). C N (0; 3; \u22125). D Q(0; 5; \u22123). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d thay \u0111\u1ed5i, song song v\u1edbi tr\u1ee5c Oz v\u00e0 c\u00e1ch tr\u1ee5c K z 3 4y Oz m\u1ed9t kho\u1ea3ng b\u1eb1ng 3 n\u00ean d n\u1eb1m tr\u00ean m\u1eb7t tr\u1ee5 tr\u00f2n xoay c\u00f3 x d A tr\u1ee5c l\u00e0 Oz v\u00e0 b\u00e1n k\u00ednh b\u1eb1ng 3. G\u1ecdi I l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a A l\u00ean Oy, kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn d O dmin nh\u1ecf nh\u1ea5t khi d \u0111i qua giao \u0111i\u1ec3m c\u1ee7a Oy v\u1edbi m\u1eb7t tr\u1ee5 l\u00e0 \u0111i\u1ec3m I(0; 3; 0) \uf8f1x = 0 \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng d : y = 3 \uf8f4\uf8f3z = t. N\u00ean d \u0111i qua \u0111i\u1ec3m N (0; 3; \u22125) A Ch\u1ecdn \u0111\u00e1p \u00e1n C 564 S\u0110T: 0905.193.688 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 102 (C\u00e2u 39 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y + 1)2 + (z + 1)2 = 9 v\u00e0 \u0111i\u1ec3m A(2; 3; \u22121). 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 A 6x + 8y + 11 = 0. B 3x + 4y + 2 = 0. C 3x + 4y \u2212 2 = 0. D 6x + 8y \u2212 11 = 0. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(\u22121; \u22121; \u2212\u221a1) v\u00e0 b\u00e1n k\u00ednh R = 3. * Ta t\u00ednh \u0111\u01b0\u1ee3c AI = 5, AM = AI2 \u2212 R2 = 4. * Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u (S ) t\u00e2m A(2; 3; \u22121), b\u00e1n k\u00ednh AM = 4 l\u00e0 (x \u2212 2)2 + (y \u2212 3)2 + (z + 1)2 = 16. * M lu\u00f4n thu\u1ed9c m\u1eb7t ph\u1eb3ng (P ) = (S) \u2229 (S ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh: 3x + 4y \u2212 2 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 103 (C\u00e2u 42 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 2)2 + (y \u2212 3)2 + (z \u2212 4)2 = 2 v\u00e0 \u0111i\u1ec3m A(1; 2; 3). X\u00e9t \u0111i\u1ec3m M thu\u1ed9c m\u1eb7t c\u1ea7u (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 + 15 = 0. B 2x + 2y + 2z \u2212 15 = 0. C x + y + z + 7 = 0. D x + y + z \u2212 7 = 0. \u0253 L\u1eddi gi\u1ea3i. MTa\u1eb7tc\u00f3c\u1ea7I#uA\u00bb(=S)(\u2212c\u00f31t;\u00e2\u2212m1;I\u2212(21;)3\u21d2; 4)I b\u00e1n k\u221a\u00ednh R = \u221a 2. A= 3. M Suy ra \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i m\u1eb7t c\u1ea7u (S). IH Do \u0111\u00f3 t\u1eadp h\u1ee3p t\u1ea5t c\u1ea3 c\u00e1c \u0111i\u1ec3m M n\u1eb1m tr\u00ean m\u1eb7t ph\u1eb3ng c\u1ed1 \u0111\u1ecbnh (\u03b1). M\u1eb7t ph\u1eb3ng c\u1ed1 v\u0111\u00e0\u1ecbnnhh(\u1ead\u03b1n)I#\u0111A\u00bbi q=ua(\u2212\u0111i1\u1ec3;m\u2212H1; l\u00e0 h\u00ecnh chi\u1ebfu A c\u1ee7a \u0111i\u1ec3m M xu\u1ed1ng IA \u22121) l\u00e0m v\u00e9c- t\u01a1 ph\u00e1p tuy\u1ebfn. Do hai tam gi\u00e1c M HI v\u00e0 AM I \u0111\u1ed3ng d\u1ea1ng n\u00ean suy ra IM 2 = IH \u00b7 IA \u21d2 IH = IM 2 = \u221a2 . IA 3 Suy ra #\u00bb = 2#\u00bb \u21d2 H \u00c54 7 10 \u00e3 IA IA ; ; . 3 33 3 M\u1eb7t ph\u1eb3ng c\u1ea7n t\u00ecm c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 \u00c5 \u2212 4\u00e3 \u2212 \u00c5 \u2212 7\u00e3 \u2212 \u00c5 \u2212 10 \u00e3 = 0 \u21d4 x + y + z \u2212 7 = 0. \u2212x y z 33 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 104 (C\u00e2u 48 - M\u0110 101 - BGD&\u0110T - N+\u0103ym2 +20\u00c41z8+-\u221a220\u00e4129)=. 3. C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau \u0111i\u1ec3m Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 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 8. C 16. D 4. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) : x2 + y2 + \u00c4 + \u221a \u00e42 = 3 c\u00f3 t\u00e2m I \u00c4 0; \u2212\u221a2\u00e4, b\u00e1n k\u00ednh R = \u221a z 2 0; 3. Ta c\u00f3 A(a; b; c) \u2208 (Oxy) \u21d2 A(a; b; 0). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 565 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian D\u1ec5 th\u1ea5y (S) c\u1eaft m\u1eb7t ph\u1eb3ng (Oxy) n\u00ean t\u1eeb m\u1ed9t \u0111i\u1ec3m A b\u1ea5t k\u1ef3 thu\u1ed9c m\u1eb7t ph\u1eb3ng (Oxy) v\u00e0 n\u1eb1m ngo\u00e0i (S) k\u1ebb ti\u1ebfp tuy\u1ebfn t\u1edbi (S) th\u00ec c\u00e1c ti\u1ebfp tuy\u1ebfn \u0111\u00f3 n\u1eb1m tr\u00ean m\u1ed9t m\u1eb7t n\u00f3n \u0111\u1ec9nh A, c\u00e1c ti\u1ebfp \u0111i\u1ec3m n\u1eb1m tr\u00ean m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh. C\u00f2n n\u1ebfu A thu\u1ed9c (S) th\u00ec ta k\u1ebb c\u00e1c ti\u1ebfp tuy\u1ebfn \u0111\u00f3 s\u1ebd thu\u1ed9c m\u1ed9t m\u1eb7t ph\u1eb3ng ti\u1ebfp di\u1ec7n c\u1ee7a (S) t\u1ea1i \u0111i\u1ec3m A. \u0110\u1ec3 c\u00f3 \u00edt nh\u1ea5t hai ti\u1ebfp tuy\u1ebfn qua A th\u1ecfa m\u00e3n b\u00e0i to\u00e1n khi v\u00e0 ch\u1ec9 khi \u221a Ho\u1eb7c A thu\u1ed9c (S) \u21d4 IA = R = 3. Ho\u1eb7c c\u00e1c ti\u1ebfp tuy\u1ebfn\u221at\u1ea1o th\u00e0nh m\u1eb7t\u221an\u00f3n v\u00e0\u221ag\u00f3c \u1edf \u0111\u221a\u1ec9nh c\u1ee7a m\u1eb7t n\u00f3n l\u00e0 M\u00f7AN \u2265 90\u25e6 \u21d4 M\u2019AI \u2265 45\u25e6. 2 IM 2 32 \u221a Suy ra sin M\u2019AI \u2265 \u21d4 \u2265 \u21d4 \u2265 \u21d4 IA \u2264 6. 2 IA 2 IA 2 \u221a\u221a V\u1eady \u0111i\u1ec1u ki\u1ec7n b\u00e0i to\u00e1n l\u00e0 3 \u2264 IA \u2264 6 \u21d4 3 \u2264 IA2 \u2264 6. Ta c\u00f3 3 \u2264 IA2 \u2264 6 \u21d4 3 \u2264 a2 + b2 + 2 \u2264 6 \u21d4 1 \u2264 a2 + b2 \u2264 4 (*). Do A(a; b; 0) c\u00f3 t\u1ecda \u0111\u1ed9 nguy\u00ean n\u00ean ta c\u00f3 \u0111i\u1ec3m th\u1ecfa m\u00e3n (\u2217) l\u00e0 (0; 2; 0), (0; \u22122; 0), (2; 0; 0), (\u22122; 0; 0), (0; 1; 0), (0; \u22121; 0), (1; 0; 0), (\u22121; 0; 0), (1; 1; 0), (1; \u22121; 0), (\u22121; 1; 0), (\u22121; \u22121; 0). V\u1eady c\u00f3 12 \u0111i\u1ec3m A th\u1ecfa m\u00e3n y\u00eau c\u1ea7u. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 105 (C\u00e2u 49 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + y2 + (z \u2212 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 v\u1edbi nhau? A 12. B 16. C 20. D 8. \u0253 L\u1eddi gi\u1ea3i. R I A (Oxy) \u221a M\u1eb7t c\u1ea7u c\u00f3 t\u00e2m I(0; 0; 1), b\u00e1n k\u00ednh R = 5. V\u00ec A \u2208 (Oxy) n\u00ean c = 0. C\u00e1c giao tuy\u1ebfn c\u1ee7a A \u0111\u1ebfn m\u1eb7t c\u1ea7u (n\u1ebfu IA > R ) t\u1ea1o n\u00ean m\u1ed9t m\u1eb7t n\u00f3n t\u221a\u00e2m A, \u0111\u1ec3 m\u1eb7t n\u00f3n n\u00e0\u221ay c\u00f3 hai \u0111\u01b0\u1eddng sinh vu\u00f4ng g\u00f3c th\u00ec g\u00f3c c\u1ee7a m\u1eb7t n\u00f3n n\u00e0y ph\u1ea3i \u2265 90\u25e6 hay IA \u2264 R 2. V\u1eady R \u2264 IA \u2264 R 2 \u21d4 5 \u2264 a2 + b2 + 1 \u2264 10 \u21d4 4 \u2264 a2 + b2 \u2264 9 Ta c\u00f3 c\u00e1c b\u1ed9 s\u1ed1 th\u1ecfa m\u00e3n (0; \u00b12) ; (0; \u00b13) ; (\u00b11; \u00b12) ; (\u00b12; \u00b12) ; (\u00b12; \u00b11) ; (\u00b12; 0) ; (\u00b13; 0), 20 b\u1ed9 s\u1ed1. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 106 (C\u00e2u 50 - M\u0110 102 - 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 \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 \u0111i\u1ec3m M c\u1eaft c\u00e1c 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 4. B 1. C 3. D 2. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 566 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. (S) c\u00f3 t\u00e2m I(2; 3; 1), b\u00e1n k\u00ednh R = 1. I B MMDD-109 Do m\u1eb7t ph\u1eb3ng (M AB) (M kh\u00f4ng tr\u00f9ng v\u1edbi A v\u00e0 B v\u00ec d(I, Ox) > 1, d(I, Oy) > 1) l\u00e0 ti\u1ebfp di\u1ec7n c\u1ee7a (S) t\u1ea1i M \u21d2 IM \u22a5 (M AB). Ta c\u00f3 IA2 = (a \u2212 2)2 + 10; IB2 = (b \u2212 3)2 + 5 \u21d2 M A2 = (a \u2212 2)2 + 9; M B2 = (b \u2212 3)2 + 4. V\u00ec A\u00f7M B = 90\u25e6 \u21d4 M A2 + M B2 = AB2 \u21d4 (a \u2212 2)2 + 9 + (b \u2212 3)2 + 4 = a2 + b2 \u21d4 2a + 3b = 13. A Do a, b \u2208 N\u2217 n\u00ean \u00f1a = 5, b=1 Suy ra c\u00f3 hai c\u1eb7p \u0111i\u1ec3m A, B. a = 2, . b=3 Th\u1eed l\u1ea1i Tr\u01b0\u1eddng h\u1ee3p 1: A(5; 0; 0) v\u00e0 B(0; 1; 0). G\u1ecdi (P ) l\u00e0 ti\u1ebfp di\u1ec7n c\u1ee7a (S) \u0111i qua A, B. Tr\u01b0\u1eddng h\u1ee3p (P ) c\u1eaft Oz t\u1ea1i C(0; 0; c), c = 0, suy ra (P ) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh (P ) : x + y + z \u2212 1 = 0. 51c (P ) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) n\u00ean 2+ 3+ 1\u22121 144 24 1 26 1 \u21d4 c = \u221260. 51c ++ + =1\u21d4 = \u20261 1 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 (P ) kh\u00f4ng c\u1eaft Oz, suy ra (P ) \u2225 Oz n\u00ean (P ) c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn #n\u00bb = \u00ee# \u00bb #j\u00bb\u00f3 = (1; 5; 0). AB, Ph\u01b0\u01a1ng tr\u00ecnh (P ) : x + 5y \u2212 5 = 0. Khi \u0111\u00f3 d(I, (P )) = |2 \u221a+ 5 \u00b7 3 \u2212 5| = 6 \u221a 26 = 1 (lo\u1ea1i). 12 + 52 13 V\u1eady tr\u01b0\u1eddng h\u1ee3p 1 c\u00f3 m\u1ed9t \u0111i\u1ec3m M th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Tr\u01b0\u1eddng h\u1ee3p 2: A(2; 0; 0) v\u00e0 B(0; 3; 0). T\u00ednh to\u00e1n t\u01b0\u01a1ng t\u1ef1, ta c\u0169ng c\u00f3 m\u1ed9t \u0111i\u1ec3m M th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. V\u1eady c\u00f3 t\u1ea5t c\u1ea3 2 \u0111i\u1ec3m M th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 107 (C\u00e2u 50 - M\u0110 104 - 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 + 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 c\u00e1c 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 4. C 1. D 3. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 567","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(3; 2; \u22121), b\u00e1n k\u00ednh R = 1. \u00c5a b \u00e3 G\u1ecdi K l\u00e0 trung \u0111i\u1ec3m AB suy ra to\u1ea1 \u0111\u1ed9 K ; ; 0 . 22 Do A\u00f7M B = 90\u25e6 n\u00ean tam\u221agi\u00e1c AM B vu\u00f4ng t\u1ea1i M n\u00ean M K = AB \u21d4 M K = a2 + b2 . 22 Do K n\u1eb1m tr\u00ean ti\u1ebfp di\u1ec7n c\u1ee7a (S) t\u1ea1i M n\u00ean M K \u22a5 IK I B \u21d2 IM2 + MK2 = IK2 M a 2 \u00c5 b \u00e32 a2 + b2 \u21d4 \u22123 + \u22122 +1=1+ 22 4 K \u21d4 3a + 2b = 13 A \u21d4 13 \u2212 3a b= . 2 Do a v\u00e0 b nguy\u00ean d\u01b0\u01a1ng v\u00e0 A, B n\u1eb1m ngo\u00e0i m\u1eb7t c\u1ea7u n\u00ean \u00ae13 \u2212 3a > 0 \u21d2 a \u2208 {1; 2; 3; 4}. a>0 Ta c\u00f3 b\u1ea3ng sau a1 2 3 4 b 5 3,5 2 0,5 T\u1eeb b\u1ea3ng tr\u00ean ta th\u1ea5y ch\u1ec9 x\u1ea3y hai ra tr\u01b0\u1eddng h\u1ee3p BB TH1: a = 1 v\u00e0 b = 5 hay A(1; 0; 0) v\u00e0 B(0; 5; 0). M1 Ta ch\u1ec9 d\u1ef1ng \u0111\u01b0\u1ee3c 2 m\u1eb7t ph\u1eb3ng ti\u1ebfp di\u1ec7n ch\u1ee9a A v\u00e0 B. TH2: a = 3 v\u00e0 b = 2 hay A (3; 0; 0) v\u00e0 B (0; 2; 0). Ta ch\u1ec9 d\u1ef1ng \u0111\u01b0\u1ee3c 2 m\u1eb7t ph\u1eb3ng ti\u1ebfp di\u1ec7n ch\u1ee9a A v\u00e0 B . L\u1ea1i c\u00f3 4 \u0111i\u1ec3m n\u00e0y n\u1eb1m tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy) v\u00e0 d[I, (Oxy)] = 1 n\u00ean (Oxy) ch\u00ednh l\u00e0 m\u1ed9t ti\u1ebfp di\u1ec7n chung. V\u1eady ch\u1ec9 c\u00f3 3 ti\u1ebfp di\u1ec7n th\u1ecfa m\u00e3n n\u00ean ch\u1ec9 c\u00f3 3 ti\u1ebfp \u0111i\u1ec3m. A A M2 M3 I Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 108 (C\u00e2u 49 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) t\u00e2m I(1; 3; 9) b\u00e1n k\u00ednh b\u1eb1ng 3. G\u1ecdi M , N l\u00e0 hai \u0111i\u1ec3m l\u1ea7n l\u01b0\u1ee3t thu\u1ed9c hai tr\u1ee5c Ox, Oz sao cho \u0111\u01b0\u1eddng th\u1eb3ng M N ti\u1ebfp x\u00fac v\u1edbi (S), \u0111\u1ed3ng th\u1eddi m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OI M N c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 13 \u00b7 G\u1ecdi A l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a MN v\u00e0 (S), gi\u00e1 tr\u1ecb AM.AN 2 b\u1eb1ng \u221a \u221a A 39. B 12 3. C 18. D 28 3. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 568 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. \u0110\u1eb7t M (a; 0; 0) v\u00e0 N (0; 0; b). Nh\u1eadn x\u00e9t: (S) ti\u1ebfp x\u00fac (Oxz) m\u00e0 M N \u2282 (Oxz) ti\u1ebfp x\u00fac (S). Suy ra M N ti\u1ebfp x\u00fac (S) t\u1ea1i ti\u1ebfp \u0111i\u1ec3m c\u1ee7a (S) v\u00e0 (Oxz) \u21d2 A(1; 0; 9). \u00aeA# M\u00bb = (a \u2212 1; 0; \u22129) \u21d2 a\u22121 = \u22129 \u21d2 (a \u2212 1)(b \u2212 9) = 9. \u22121 b\u22129 #\u00bb = (\u22121; 0; b \u2212 9) AN Khi \u0111\u00f3 OIM N c\u00f3 OM N vu\u00f4ng t\u1ea1i O, (IM N ) \u22a5 (OM N ) (do IA \u2282 (IM N ), IA \u22a5 (OM N )). Suy ra B\u00e1n k\u00ednh m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp OIM N b\u1eb1ng b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp IM N b\u1eb1ng 13 \u00b7 2 1 \u00b7 3.M N IM.IN.M N \u21d4 IM.IN Suy ra 2 = = 39. (1) 4. 13 2 M\u00e0 IM = (a \u2212 1)2 + 32 + 92 = (a \u2212 1)2 + 90. \u00a0 81 \u00b7 IN = 12 + 32 + (b \u2212 9)2 = 10 + (a \u2212 1)2 \u2212 1)2 \u00ee 81 \u00f3 \u21d4 \u2212 1)2 10 Thay v\u00e0o (1) ta \u0111\u01b0\u1ee3c: [(a + 90] + \u221a(a\u22121)2 = 1521 (a = 27. \uf8f1 \u00bb \u221a \uf8f2AM = (a \u2212 1)2 + 81 = 108 = 6 3 \u221a Ta c\u00f3 \u221a , suy ra AM.AN = 12 3. \u00bb \uf8f3AN = 1 + (b \u2212 9)2 = 1 + 3 = 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 109 (C\u00e2u 41 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(\u22121; 2; 1) v\u00e0 \u0111i qua \u0111i\u1ec3m A(1; 0; \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 l\u1edbn nh\u1ea5t b\u1eb1ng A 64 B 32. C 64. D 32 . . 3 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t AD = a, AB = b, AC = c. D A Khi \u0111\u00f3, VABCD = 1AB \u00b7 AC \u00b7 AD = 1 \u221a 6 abc. 6 Ta c\u00f3 b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u (S) l\u00e0 R = IA = 2 3. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m BC. Khi \u0111\u00f3, AM = b2 + c2 . 2 V\u00ec t\u1ee9 di\u1ec7n ABCD n\u1ed9i ti\u1ebfp trong m\u1eb7t c\u1ea7u (S) n\u00ean ta c\u00f3 IM \u2225 AD I B 11 v\u00e0 IM = AD = a. M 22 X\u00e9t tam gi\u00e1c AIM vu\u00f4ng t\u1ea1i M , ta c\u00f3 AI2 = AM 2 + IM 2 \u21d4 a2 + b2 + c2 = 48 C Suy ra VA2BCD = 1 a2b2c2 \u2264 1 (a2 + b2 + c2)3 = 1024 hay VABCD \u2264 32 36 36 27 9 . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 110 (C\u00e2u 37 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m E(2; 1; 3), m\u1eb7t ph\u1eb3ng (P ) : 2x + 2y \u2212 z \u2212 3 = 0 v\u00e0 m\u1eb7t c\u1ea7u (S) : (x \u2212 3)2 + (y \u2212 2)2 + (z \u2212 5)2 = 36. G\u1ecdi \u2206 l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua E, n\u1eb1m trong (P ) v\u00e0 c\u1eaft (S) t\u1ea1i hai \u0111i\u1ec3m c\u00f3 kho\u1ea3ng c\u00e1ch nh\u1ecf nh\u1ea5t. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a \u2206 l\u00e0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 569 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian \uf8f1x = 2 + 9t \uf8f1x = 2 \u2212 5t \uf8f1x = 2 + t \uf8f1x = 2 + 4t \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f2 \uf8f2 \uf8f2 A y = 1 + 9t . B y = 1 + 3t . C y = 1\u2212t. D y = 1 + 3t . \uf8f3\uf8f4z = 3 + 8t \uf8f3\uf8f4z = 3 \uf8f4\uf8f3z = 3 \uf8f3\uf8f4z = 3 \u2212 3t \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7\u221au (S) c\u00f3 t\u00e2m I (\u221a3; 2; 5) v\u00e0 b\u00e1n k\u00ednh R = 6. IE = 12 + 12 + 22 = 6 < R, suy ra \u0111i\u1ec3m E n\u1eb1m trong m\u1eb7t c\u1ea7u (S). G\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a I tr\u00ean m\u1eb7t ph\u1eb3ng (P ), A v\u00e0 B l\u00e0 hai giao \u0111i\u1ec3m c\u1ee7a \u2206 v\u1edbi (S). Khi \u0111\u00f3, AB nh\u1ecf nh\u1ea5t \u21d4 d(J, \u2206) l\u1edbn nh\u1ea5t (v\u1edbi J l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n giao tuy\u1ebfn c\u1ee7a (P ) v\u00e0 (S)), m\u00e0 d(J, \u2206) \u2264 EJ. Do \u0111\u00f3 AB nh\u1ecf nh\u1ea5t \u21d4 AB \u22a5 OE, m\u00e0 AB \u22a5 IH n\u00ean AB \u22a5 (HIE) \u21d2 AB \u22a5 IE. u# \u2206\u00bb \u00een# P\u00bb; # \u00bb\u00f3 Suy ra: = EI = (5; \u22125; 0) = 5 (1; \u22121; 0). \uf8f1x = 2 + t \uf8f4 \uf8f2 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a \u2206 l\u00e0 y = 1 \u2212 t . \uf8f3\uf8f4z = 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 111 (C\u00e2u 42 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(0; 3; \u22122). X\u00e9t \u0111\u01b0\u1eddng th\u1eb3ng d thay \u0111\u1ed5i song song v\u1edbi Oz v\u00e0 c\u00e1ch Oz m\u1ed9t kho\u1ea3ng b\u1eb1ng 2. Khi kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn d nh\u1ecf nh\u1ea5t th\u00ec d \u0111i qua \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A P (\u22122; 0; \u22122). B N (0; \u22122; \u22125). C Q(0; 2; \u22125). D M (0; 4; \u22122). \u0253 L\u1eddi gi\u1ea3i. V\u00ec d song song v\u1edbi Oz v\u00e0 c\u00e1ch Oz m\u1ed9t kho\u1ea3ng b\u1eb1ng 2 n\u00ean d thu\u1ed9c Z m\u1eb7t tr\u1ee5 tr\u1ee5c Oz v\u00e0 b\u00e1n k\u00ednh b\u1eb1ng 2. 2d C\u00f3 HH# A(\u00bb0;=0;(\u22120;23); l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A(0; 3; \u22122) tr\u00ean Oz. A C\u00f3 0) \u21d2 HA = 3 n\u00ean A n\u1eb1m ngo\u00e0i m\u1eb7t tr\u1ee5. H G\u1ecdi (P) l\u00e0 m\u1eb7t ph\u1eb3ng qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi Oz. K M l\u00e0 \u0111i\u1ec3m tr\u00ean d. M G\u1ecdi K l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AH v\u00e0 m\u1eb7t tr\u1ee5 (K n\u1eb1m gi\u1eefa A v\u00e0 H). O D\u1ec5 th\u1ea5y AM \u2265 AK; AK = AH \u2212 d(OZ; d) = 1 = d(A; d). D\u1ea5u b\u1eb1ng x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi M \u2261 K. Khi \u0111\u00f3 ta c\u00f3 H# K\u00bb = 2H# A\u00bb \u21d2 K(0; 2; \u22122). 3 \uf8f1x = 0 \uf8f4 \uf8f2 \u21d2 d: y = 2 (t \u2208 R). \uf8f4\uf8f3z = \u22122 + t V\u1edbi t = \u22123 ta th\u1ea5y d \u0111i qua \u0111i\u1ec3m Q. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 112 (C\u00e2u 45 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(0; 3; \u22122). X\u00e9t \u0111\u01b0\u1eddng th\u1eb3ng d thay \u0111\u1ed5i, song song v\u1edbi tr\u1ee5c Oz v\u00e0 c\u00e1ch tr\u1ee5c Oz m\u1ed9t kho\u1ea3ng b\u1eb1ng 2. Khi kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn d l\u1edbn nh\u1ea5t, d \u0111i qua \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A Q(\u22122; 0; \u22123). B M (0; 8; \u22125). C N (0; 2; \u22125). D P (0; \u22122; \u22125). \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 570 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN d Oz BH A(0; 3; \u22122) Do \u0111\u01b0\u1eddng th\u1eb3ng d \u2225 Oz n\u00ean d n\u1eb1m tr\u00ean m\u1eb7t tr\u1ee5 c\u00f3 tr\u1ee5c l\u00e0 Oz v\u00e0 b\u00e1n k\u00ednh tr\u1ee5 l\u00e0 R = 2. G\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a A tr\u00ean tr\u1ee5c Oz, suy ra t\u1ecda \u0111\u1ed9 H(0; 0; \u22122). Do \u0111\u00f3 d(A, Oz) = AH = 3. G\u1ecdi B l\u00e0 \u0111i\u1ec3m thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng AH sao cho #\u00bb = 3# \u00bb Suy ra B(0; \u22122; \u22122). AH AB. 5 V\u1eady d(A, d)max = 5 \u21d4 d l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua B v\u00e0 song song v\u1edbi Oz. \uf8f1x = 0 \uf8f4 \uf8f2 Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a d : y = \u22122 \uf8f4\uf8f3z = \u22122 + t. K\u1ebft lu\u1eadn: d \u0111i qua \u0111i\u1ec3m P (0; \u22122; \u22125). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 113 (C\u00e2u 49 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(1; \u22123; \u22124) v\u00e0 B(\u22122; 1; 2). X\u00e9t hai \u0111i\u1ec3m M v\u00e0 N thay \u0111\u1ed5i thu\u1ed9\u221ac m\u1eb7t ph\u1eb3ng (Oxy) sa\u221ao cho M N = 2. Gi\u00e1 tr\u1ecb \u221al\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN\u221a| b\u1eb1ng A 3 5. B 61. C 13. D 53. \u0253 L\u1eddi gi\u1ea3i. A1 A2 B H K NM Oxy A V\u00ec zA \u00b7 zB < 0 n\u00ean A, B n\u1eb1m kh\u00e1c ph\u00eda so v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy). G\u1ecdi H, K l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh 1ch; 0i\u1ebf)u\u21d2vuH#\u00f4nK\u00bbg g\u00f3c c\u1ee7a A, B l\u00ean m\u1eb7t ph\u1eb3ng (Oxy). Suy ra H(1; \u22123; 0), K(\u22122; = (\u22123; 4; 0) v\u00e0 HK = 5. G\u1ecdi A1 l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng c\u1ee7# a A\u00bb q=uaM# (NO\u00bbx\u21d2y) \u21d2 A1(1; \u22123; 4). G\u1ecdi A2 l\u00e0 \u0111i\u1ec3m th\u1ecfa m\u00e3n A1A2 A1A2 = 2. Do \u0111\u00f3 A2 thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n (C) n\u1eb1m trong m\u1eb7t ph\u1eb3ng song song v\u1edbi (Oxy) v\u00e0 c\u00f3 t\u00e2m A1, b\u00e1n k\u00ednh R = 2. Khi \u0111\u00f3 |AM \u2212 BN | = |A1M \u2212 BN | = |A2N \u2212 B#N | \u00bb\u2264 A2B. #\u00bb D\u1ea5u \u201c=\u201d x\u1ea3y ra v\u00e0 A2B \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t khi A1A2 ng\u01b0\u1ee3c h\u01b0\u1edbng v\u1edbi HK. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 571 S\u0110T: 0905.193.688","3. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian A# 1A\u00bb2 \u2212 A1A2 H# K\u00bb \u00c5 6 8 \u00e3 \u00c5 11 23 \u00e3 \u221a HK 0 4 53. Suy ra = \u00b7 = ; \u2212 ; \u21d2 A2 ; \u2212 ; \u21d2 A2B = 55 \u221a 55 V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN | b\u1eb1ng 53. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 114 (C\u00e2u 47 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho ba \u0111i\u1ec3m A(\u22122; 0; 0), B(0; \u22122; 0) v\u00e0 C(0; 0; \u22122). G\u1ecdi D l\u00e0 \u0111i\u1ec3m kh\u00e1c O sao cho DA, DB, DC \u0111\u00f4i m\u1ed9t vu\u00f4ng g\u00f3c v\u1edbi nhau v\u00e0 I(a; b; c) l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n ABCD.T\u00ednh S = a + b + c. A S = \u22124. B S = \u22121. C S = \u22122. D S = \u22123. \u0253 L\u1eddi gi\u1ea3i. Nh\u1eadn x\u00e9t OA = OB = OC v\u00e0 OA, OB, OC \u0111\u00f4i m\u1ed9t vu\u00f4ng g\u00f3c. C Do \u0111\u00f3 ta c\u00f3 th\u1ec3 x\u00e9t tr\u01b0\u1eddng h\u1ee3p D l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi O qua ABC. Khi \u0111\u00f3 D \u00c5 4 , \u22124, \u2212 4 \u00e3 \u2212 . T\u1eeb M# \u00bbI = D# E\u00bb3, 3 3 M I d\u1eabn \u0111\u1ebfn I \u00c5 1 , \u22121,\u22121\u00e3. E \u2212 D 333 A B V\u1eady S = \u22121. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 115 (C\u00e2u 28 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho h\u00ecnh ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh ch\u1eef nh\u1eadt, AB = a, BC = 2a, SA vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng \u221a\u0111\u00e1y v\u00e0 SA = a. Kho\u1ea3ng c\u221a\u00e1ch gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3\u221ang BD v\u00e0 SC b\u1eb1ng \u221a 30a 4 21a C2 21a 30a A . B . . D . 6 21 21 12 \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 nh\u01b0 h\u00ecnh v\u1ebd, ta c\u00f3 A(0; 0; 0), B(0; a; 0), z D(2a; 0; 0), C(2a; a; 0) v\u00e0 S(0; 0; a). Ta c\u00f3 S B# D\u00bb = (2a; \u2212a; 0). A #\u00bb = (2a; a; \u2212a). SC S# B\u00bb = (0; a; \u2212a). B [B# D\u00bb, S# C\u00bb] = (a2; 2a2\u221a; 4a2) y \u21d2 # \u00bb #\u00bb a2 21. [BD, SC] = C [B# D\u00bb, S# C\u00bb] \u00b7 S# B\u00bb = 2a3. D x Kho\u1ea3ng c\u00e1ch gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng BD v\u00e0 SC l\u00e0 [B# D\u00bb, S# C\u00bb] \u00b7 S# B\u00bb \u221a 2a 21 d(SC, BD) = =. [B# D\u00bb, S# C\u00bb] 21 Ch\u1ecdn \u0111\u00e1p \u00e1n C 572 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t"]