SÁCH EBOOK HÌNH HỌC 12

["6. Trong kh\u00abng gian Oxyz cho m\u00c6t c\u00c7u (S) c\u00e3 ph\u2212\u00acng tr\u00d7nh x2 \uf02b y2 \uf02b z2 \uf03d 4a2 (a > 0). a) T\u00ddnh di\u00d6n t\u00ddch m\u00c6t c\u00c7u (S) v\u03bc th\u00d3 t\u00ddch c\u00f1a kh\u00e8i c\u00c7u t\u2212\u00acng \u00f8ng. b) M\u00c6t c\u00c7u (S) c\u00bet m\u00c6t ph\u00bcng (Oxy) theo \u00ae\u2212\u00eang tr\u00dfn (C). X\u00b8c \u00ae\u00denh t\u00a9m v\u03bc b\u00b8n k\u00ddnh c\u00f1a (C). c) T\u00ddnh di\u00d6n t\u00ddch xung quanh c\u00f1a h\u00d7nh tr\u00f4 nh\u00cbn (C) l\u03bcm \u00ae\u00b8y v\u03bc c\u00e3 chi\u00d2u cao l\u03bc a 3. T\u00ddnh th\u00d3 t\u00ddch c\u00f1a kh\u00e8i tr\u00f4 t\u2212\u00acng \u00f8ng. 7. Trong kh\u00abng gian Oxyz cho hai \u00ae\u2212\u00eang th\u00bcng d1 v\u03bc d2 c\u00e3 ph\u2212\u00acng tr\u00d7nh \uf0ecx \uf03d1\uf02dt \uf0ecx \uf03d 2t\uf0a2 \uf0ef v\u03bc d2 : \uf0ed\uf0efy \uf03d \uf02d1 \uf02b t\uf0a2 d1 : \uf0ed y \uf03d t \uf0ee\uf0efz \uf03d t\uf0a2. \uf0ee\uf0efz \uf03d \uf02dt a) Ch\u00f8ng minh r\u00bbng hai \u00ae\u2212\u00eang th\u00bcng d1 v\u03bc d2 ch\u00d0o nhau. b) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (\uf061) ch\u00f8a d1 v\u03bc song song v\u00edi d2. 8. Trong kh\u00abng gian Oxyz cho c\u00b8c \u00aei\u00d3m A(1 ; 0 ; \u20131), B(3 ; 4 ; \u20132), C(4 ; \u20131 ; 1), D(3 ; 0 ; 3). a) Ch\u00f8ng minh r\u00bbng A, B, C, D kh\u00abng \u00ae\u00e5ng ph\u00bcng. b) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (ABC) v\u03bc t\u00ddnh kho\u00b6ng c\u00b8ch t\u00f5 D \u00ae\u00d5n (ABC). c) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t c\u00c7u ngo\u00b9i ti\u00d5p t\u00f8 di\u00d6n ABCD. d) T\u00ddnh th\u00d3 t\u00ddch t\u00f8 di\u00d6n ABCD. 9. Trong kh\u00abng gian Oxyz cho b\u00e8n \u00aei\u00d3m A(2 ; 4 ; \uf02d1), B(1 ; 4 ; \uf02d1), C(2 ; 4 ; 3), D(2 ; 2 ; \uf02d1). a) Ch\u00f8ng minh r\u00bbng c\u00b8c \u00ae\u2212\u00eang th\u00bcng AB, AC, AD vu\u00abng g\u00e3c v\u00edi nhau t\u00f5ng \u00ae\u00abi m\u00e9t. T\u00ddnh th\u00d3 t\u00ddch kh\u00e8i t\u00f8 di\u00d6n ABCD. b) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t c\u00c7u (S) \u00aei qua b\u00e8n \u00aei\u00d3m A, B, C, D. c) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (\uf061) ti\u00d5p x\u00f3c v\u00edi m\u00c6t c\u00c7u (S) v\u03bc song song v\u00edi m\u00c6t ph\u00bcng (ABD). \uf0ecx \uf03d 1\uf02d 2t 10. Trong kh\u00abng gian Oxyz cho \u00ae\u2212\u00eang th\u00bcng d : \uf0ed\uf0efy \uf03d 2 \uf02b t \uf0ef\uf0eez \uf03d 3 \uf02d t v\u03bc m\u00c6t ph\u00bcng (\uf061) : 2x + y + z = 0. 100","a) T\u00d7m to\u00b9 \u00ae\u00e9 giao \u00aei\u00d3m A c\u00f1a d v\u03bc (\uf061). b) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (\uf062) qua A v\u03bc vu\u00abng g\u00e3c v\u00edi d. 11. Trong kh\u00abng gian Oxyz cho c\u00b8c \u00aei\u00d3m A(\uf02d1 ; 2 ; 0), B(\uf02d3 ; 0 ; 2), C(1 ; 2 ; 3), D(0 ; 3 ; \uf02d2). a) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (ABC) v\u03bc ph\u2212\u00acng tr\u00d7nh tham s\u00e8 c\u00f1a \u00ae\u2212\u00eang th\u00bcng AD. b) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (\uf061) ch\u00f8a AD v\u03bc song song v\u00edi BC. 12. Trong kh\u00abng gian Oxyz cho b\u00e8n \u00aei\u00d3m A(3 ; \uf02d2 ; \uf02d2), B(3 ; 2 ; 0), C(0 ; 2 ; 1) v\u03bc D(\uf02d1 ; 1 ; 2) a) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (BCD). Suy ra ABCD l\u03bc m\u00e9t t\u00f8 di\u00d6n. b) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t c\u00c7u (S) t\u00a9m A v\u03bc ti\u00d5p x\u00f3c v\u00edi m\u00c6t ph\u00bcng (BCD). c) T\u00d7m to\u00b9 \u00ae\u00e9 ti\u00d5p \u00aei\u00d3m H c\u00f1a (S) v\u03bc m\u00c6t ph\u00bcng (BCD). 13. Trong kh\u00abng gian Oxyz, cho hai \u00ae\u2212\u00eang th\u00bcng : \uf0ecx \uf03d \uf02d1 \uf02b 3t \uf0ecx \uf03d t\uf0a2 d1 : \uf0ed\uf0efy \uf03d 1 \uf02b 2t v\u03bc d2 : \uf0ef\uf0edy \uf03d 1 \uf02b t\uf0a2 \uf0ee\uf0efz \uf03d 3 \uf02d 2t \uf0ef\uf0eez \uf03d \uf02d3 \uf02b 2t\uf0a2. a) Ch\u00f8ng minh d1 v\u03bc d2 c\u00efng thu\u00e9c m\u00e9t m\u00c6t ph\u00bcng. b) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng \u00ae\u00e3. 14. Trong kh\u00abng gian cho ba \u00aei\u00d3m A, B, C. \uf075\uf075\uf072 \uf075\uf075\uf072 \uf075\uf075\uf075\uf072 \uf072 a) X\u00b8c \u00ae\u00denh \u00aei\u00d3m G sao cho GA \uf02b 2GB \uf02d 2GC \uf03d 0 . b) T\u00d7m t\u00cbp h\u00eep c\u00b8c \u00aei\u00d3m M sao cho MA2 + 2MB2 \u2013 2MC2 = k2, v\u00edi k l\u03bc h\u00bbng s\u00e8. 15. Cho hai \u00ae\u2212\u00eang th\u00bcng ch\u00d0o nhau \uf0ecx \uf03d 2\uf02dt \uf0ecx \uf03d 2 \uf02b 2t\uf0a2 d : \uf0ed\uf0efy \uf03d \uf02d1 \uf02b t v\u03bc d' : \uf0ef\uf0edy \uf03d t\uf0a2 \uf0ef\uf0eez \uf03d 1 \uf02d t \uf0ee\uf0efz \uf03d 1 \uf02b t\uf0a2. a) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh c\u00b8c m\u00c6t ph\u00bcng (\uf061) v\u03bc (\uf062) song song v\u00edi nhau v\u03bc l\u00c7n l\u2212\u00eet ch\u00f8a d v\u03bc d'. 101","b) L\u00cay hai \u00aei\u00d3m M(2 ; \uf02d1 ; 1) v\u03bc M'(2 ; 0 ; 1) l\u00c7n l\u2212\u00eet tr\u00aan d v\u03bc d'. T\u00ddnh kho\u00b6ng c\u00b8ch t\u00f5 M \u00ae\u00d5n m\u00c6t ph\u00bcng (\uf062) v\u03bc kho\u00b6ng c\u00b8ch t\u00f5 M' \u00ae\u00d5n m\u00c6t ph\u00bcng (\uf061). So s\u00b8nh hai kho\u00b6ng c\u00b8ch \u00ae\u00e3. 16. Trong kh\u00abng gian Oxyz cho m\u00c6t ph\u00bcng (\uf061) c\u00e3 ph\u2212\u00acng tr\u00d7nh 4x + y + 2z + 1 = 0 v\u03bc m\u00c6t ph\u00bcng (\uf062) c\u00e3 ph\u2212\u00acng tr\u00d7nh 2x \u2013 2y + z + 3 = 0. a) Ch\u00f8ng minh r\u00bbng (\uf061) c\u00bet (\uf062). b) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh tham s\u00e8 c\u00f1a \u00ae\u2212\u00eang th\u00bcng d l\u03bc giao c\u00f1a (\uf061) v\u03bc (\uf062). c) T\u00d7m \u00aei\u00d3m M' \u00ae\u00e8i x\u00f8ng v\u00edi \u00aei\u00d3m M(4 ; 2 ; 1) qua m\u00c6t ph\u00bcng (\uf061). d) T\u00d7m \u00aei\u00d3m N' \u00ae\u00e8i x\u00f8ng v\u00edi \u00aei\u00d3m N(0 ; 2 ; 4) qua \u00ae\u2212\u00eang th\u00bcng d. 102","H\u2212\u00edng d\u00c9n gi\u00b6I b\u03bci t\u00cbp v\u03bc \u00a7\u00b8p s\u00e8 Ch\u2212\u00acng I. Kh\u00e8i \u00aea di\u00d6n \u00abn t\u00cbp ch\u2212\u00acng I \u00a71. Kh\u00b8i ni\u00d6m v\u00d2 kh\u00e8i \u00aea di\u00d6n 5. OH \uf03d abc \uf0d7 a2b2 \uf02b b2c2 \uf02b c2a2 1. v\u03bc 2. S\u00f6 d\u00f4ng t\u00ddnh ch\u00cat : M\u00e7i c\u00b9nh c\u00f1a m\u00e9t \u00aea di\u00d6n l\u03bc c\u00b9nh chung c\u00f1a \u00ae\u00f3ng hai m\u00c6t. 6. a) T\u00d8 s\u00e8 th\u00d3 t\u00ddch c\u00c7n t\u00d7m l\u03bc 5 ; 8 3. Chia h\u00d7nh l\u00cbp ph\u2212\u00acng ABCD.A'B'C'D' th\u03bcnh n\u00a8m t\u00f8 di\u00d6n : AB'CD', A'AB'D', b) VS.DBC \uf03d a3 5 3\uf0d7 BACB', C'B'CD', DACD'. 96 4. Chia h\u00d7nh l\u00cbp ph\u2212\u00acng th\u03bcnh hai l\u00a8ng tr\u00f4 7. VS.ABC \uf03d 8 3.a3 b\u00bbng nhau r\u00e5i chia m\u00e7i l\u00a8ng tr\u00f4 th\u03bcnh ba t\u00f8 di\u00d6n b\u00bbng nhau. abc5(a2 \uf02b b2 \uf02b 2c2 ) \u00a72. Kh\u00e8i \u00aea di\u00d6n l\u00e5i v\u03bc kh\u00e8i \u00aea di\u00d6n \u00ae\u00d2u 8. V \uf03d 6(a2 \uf02b c2 )(b2 \uf02b c2 )(a2 \uf02b b2 \uf02b c2 ) \uf0d7 2. T\u00d8 s\u00e8 \u00ae\u00e3 b\u00bbng 2 3. 9. V \uf03d a3 6 18 3. G\u00e4i (H) l\u03bc h\u00d7nh t\u00f8 di\u00d6n \u00ae\u00d2u c\u00b9nh a. Khi \u00ae\u00e3 t\u00a9m c\u00f1a c\u00b8c m\u00c6t c\u00f1a (H) t\u00b9o th\u03bcnh m\u00e9t t\u00f8 10. b) V \uf03d 5a3 \uf0d7 di\u00d6n (H') c\u00e3 s\u00b8u c\u00b9nh \u00ae\u00d2u b\u00bbng a \uf0d7 18 3 3 11. T\u00d8 s\u00e8 th\u00d3 t\u00ddch c\u00f1a ch\u00f3ng b\u00bbng 1. 4. \u00a7\u00d3 \u00fd r\u00bbng B, C, D, E c\u00b8ch \u00ae\u00d2u A v\u03bc F n\u00aan ch\u00f3ng \u00ae\u00e5ng ph\u00bcng. 12. a) VADMN \uf03d a3 ; 6 \u00a73. Kh\u00b8i ni\u00d6m v\u00d2 th\u00d3 t\u00ddch c\u00f1a kh\u00e8i \u00aea di\u00d6n b) T\u00d8 s\u00e8 th\u00d3 t\u00ddch ph\u00b6i t\u00d7m l\u03bc 55 \uf0d7 89 1. a3 2 \uf0d7 Ch\u2212\u00acng II. M\u00c6t n\u00e3n, m\u00c6t tr\u00f4, m\u00c6t c\u00c7u 12 \u00a71. Kh\u00b8i ni\u00d6m v\u00d2 m\u00c6t tr\u00dfn xoay 2. a3 2 \uf0d7 3 2. a) H\u00d7nh tr\u00f4 ; b) H\u00d7nh n\u00e3n ; 3. T\u00d8 s\u00e8 c\u00f1a th\u00d3 t\u00ddch l\u03bc 3. c) Kh\u00e8i n\u00e3n ; d) Kh\u00e8i tr\u00f4. 4. T\u00ddnh di\u00d6n t\u00ddch tam gi\u00b8c theo hai c\u00b9nh v\u03bc g\u00e3c xen gi\u00f7a. 3. a) Sxq \uf0bb 2514,5 cm2 ; b) V \uf020\uf0bb 13 089,969 cm3 ; 5. VD.CEF \uf03d a3 \uf0d7 c) 500 cm2. 36 4. M\u00c6t n\u00e3n nh\u00cbn AB l\u03bcm tr\u00f4c, g\u00e3c \u00eb \u00ae\u00d8nh 6. G\u00e4i h l\u03bc \u00ae\u00e9 d\u03bci \u00ae\u2212\u00eang vu\u00abng g\u00e3c chung b\u00bbng 60o. c\u00f1a d v\u03bc d', \uf061 l\u03bc g\u00e3c gi\u00f7a hai \u00ae\u2212\u00eang th\u00bcng d v\u03bc d'. Khi \u00ae\u00e3 VABCD \uf03d 1 hab.sin\uf061 . 6 103","5. a) Sxq \uf0bb 219, 91cm2 ; V \uf0bb 549,77 cm3 ; 7. a) 1 a2 \uf02b b2 \uf02b c2 ; 2 b) 56 cm2. b) 1 b2 \uf02b c2 . 6. Sxq \uf03d 2\uf070 a2 ; V \uf03d \uf070 a3 3 \uf0d7 2 3 10. S \uf03d \uf070 (a2 \uf02b b2 \uf02b c2 ) ; 7. a) Sxq \uf03d 2 3\uf070 r2 ; Stp \uf03d 2( 3 \uf02b 1)\uf070 r2 ; V \uf03d 1 \uf070 (a2 \uf02b b2 \uf02b c2 ). a2 \uf02b b2 \uf02b c2 . b) V \uf03d 3\uf070 r3 ; 6 c) r 3 \uf0d7 \u00a4n t\u00cbp ch\u2212\u00acng II 2 1. C\u00a9u a) v\u03bc d) \u00ae\u00f3ng. 8. a) 3 ; V \uf03d \uf070 a3 \uf0d7 3 b) 1 \uf0d7 2. Sxq \uf03d \uf070 a2 2; 2 5. a) AH \uf03d a 6 ; 9. a) Sxq \uf03d 2\uf070 a2 ; S\u00ae\u00b8y \uf03d \uf070 a2 ; 3 2 2 2\uf070 a2 2 ; V \uf03d \uf070 a3 6\uf0d7 V \uf03d 2\uf070 a3 ; b) Sxq \uf03d 3 9 12 6. r \uf03d 3a ; S \uf03d 9\uf070 a2 ; V \uf03d 9\uf070 a3 \uf0d7 b) S\uf056SBC \uf03d a2 2 \uf0d7 4 4 16 3 7. b) 3 \uf0d7 10. SABCD \uf03d 5r2 ; cos\uf061 \uf03d 3\uf0d7 2 2 5 \u00a72. M\u00c6t c\u00c7u Ch\u2212\u00acng III. Ph\u2212\u00acng ph\u00b8p to\u00b9 \u00ae\u00e9 trong kh\u00abng gian 1. T\u00cbp h\u00eep c\u00b8c \u00aei\u00d3m M lu\u00abn nh\u00d7n AB c\u00e8 \u00ae\u00denh d\u2212\u00edi m\u00e9t g\u00e3c vu\u00abng l\u03bc m\u00c6t c\u00c7u \u00a71. H\u00d6 to\u00b9 \u00ae\u00e9 trong kh\u00abng gian \u00ae\u2212\u00eang k\u00ddnh AB. 1. a) \uf072 \uf03d \uf0e6\uf0e8\uf0e711 ;\uf020\uf0201 ; \uf02018 1 \uf0f6 ; 2. r \uf03d a 2 \uf0d7 d 3 3 \uf0f8\uf0f7 2 \uf072 3. T\u00cbp h\u00eep t\u00a9m c\u00b8c m\u00c6t c\u00c7u lu\u00abn ch\u00f8a m\u00e9t b) e = (0 ; \uf02d27 ; 3). \u00ae\u2212\u00eang tr\u00dfn c\u00e8 \u00ae\u00denh cho tr\u2212\u00edc l\u03bc m\u00e9t \u00ae\u2212\u00eang th\u00bcng \uf044 vu\u00abng g\u00e3c v\u00edi m\u00c6t ph\u00bcng 2. G \uf03d \uf0e6 2 ; 0 ; 4 \uf0f6\uf0f8\uf0f7. ch\u00f8a \u00ae\u2212\u00eang tr\u00dfn \u00ae\u00e3 t\u00b9i t\u00a9m c\u00f1a \u00ae\u2212\u00eang \uf0e8\uf0e7 3 3 tr\u00dfn. 3. A' = (3 ; 5 ; \u20136), B' = (4 ; 6 ; \u20135) , 4. T\u00cbp h\u00eep t\u00a9m nh\u00f7ng m\u00c6t c\u00c7u c\u00efng ti\u00d5p x\u00f3c v\u00edi ba c\u00b9nh c\u00f1a m\u00e9t tam gi\u00b8c cho C =\uf072(\uf0722 ; 0 ; 2), D' = (3 ; 4 ; \u20136). tr\u2212\u00edc l\u03bc tr\u00f4c c\u00f1a \u00ae\u2212\u00eang tr\u00dfn n\u00e9i ti\u00d5p tam 4. a) a\uf072.b\uf072 = 6 ; gi\u00b8c \u00ae\u00b7 cho. b) c.d = \u201321. 5. b) d2 \u2013 r2. 104","5. a) M\u00c6t c\u00c7u t\u00a9m O = (4 ; 1 ; 0), c\u00e3 b\u00b8n \u00a73. Ph\u2212\u00acng tr\u00d7nh \u00ae\u2212\u00eang th\u00bcng trong k\u00ddnh r = 4 ; kh\u00abng gian b) M\u00c6t c\u00c7u t\u00a9m I = \uf0e6\uf0e8\uf0e71 ; \uf02d 4 ;\uf020 \uf02d 5 \uf0f6 , c\u00e3 \uf0ecx \uf03d 5 \uf02b 2t \uf0ecx \uf03d 2 \uf02b t 3 2 \uf0f8\uf0f7 1. a) d : \uf0ed\uf0efy \uf03d 4 \uf02d 3t ; \uf0ef b\u00b8n k\u00ddnh r = 3 1 \uf0d7 \uf0ee\uf0efz \uf03d 1 \uf02b t b) d : \uf0ed y \uf03d \uf02d1 \uf02b t ; 6 \uf0ecx \uf03d 2 \uf02b 2t c) d : \uf0ef\uf0edy \uf03d 3t ; \uf0ef\uf0eez \uf03d 3 \uf02d t 6. a) M\u00c6t c\u00c7u c\u00e3 ph\u2212\u00acng tr\u00d7nh l\u03bc : \uf0ef\uf0eez \uf03d \uf02d3 \uf02b 4t (x \u2013 3)2 + (y + 1)2 + (z \u2013 5)2 = 9 ; \uf0ecx \uf03d 1\uf02b 4t d) d : \uf0ed\uf0efy \uf03d 2 \uf02b 2t b) M\u00c6t c\u00c7u c\u00e3 ph\u2212\u00acng tr\u00d7nh l\u03bc : (x \u2013 3)2 + (y + 3)2 + (z \u2013 1)2 = 5. \uf0ee\uf0efz \uf03d 3 \uf02b t. \u00a72. Ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng \uf0ecx \uf03d 2 \uf02b t \uf0ecx \uf03d 0 2. a) d' : \uf0ef\uf0edy \uf03d \uf02d3 \uf02b 2t ; b) d'' : \uf0ed\uf0efy \uf03d \uf02d3 \uf02b 2t 1. a) 2x + 3y + 5z \u2013 16 = 0 ; b) x \u2013 3y + 3z \u2013 9 = 0 ; \uf0ef\uf0eez \uf03d 0 \uf0ee\uf0efz \uf03d 1 \uf02b 3t. c) 2x + 3y + 6z + 6 = 0. 3. a) d c\u00bet d' ; b) d \/\/ d'. 4. a = 0. 2. x \u2013 y \u2013 2z + 9 = 0. 5. a) 1 \u00aei\u00d3m chung ; b) 0 \u00aei\u00d3m chung ; 3. a) z = 0 ; x = 0 ; y = 0 ; c) v\u00ab s\u00e8 \u00aei\u00d3m chung. b) z = \u20133 ; x = 2 ; y = 6. 4. a) 2y + z = 0 ; b) 3x + z = 0 ; 6. d( \uf044 , (\uf061)) = 2 \uf0d7 c) 4x + 3y = 0. 3 5. a) (ACD) : 2x + y + z \u2013 14 = 0 ; 7. a) H \uf0e6 3 ;0 ; \uf02d 1 \uf0f6 ; (BCD) : 6x + 5y + 3z \u2013 42 = 0 ; \uf0e7\uf0e8 2 2 \uf0f8\uf0f7 b) (\uf061) : 10x + 9y + 5z \u2013 74 = 0. b) A'(2 ; 0 ; \u20131). 6. (\uf061) : 2x \u2013 y + 3z \u2013 11 = 0. 8. a) H(\u20131 ; 2 ; 0) ; 7. (\uf061) : x \u2013 2z + 1 = 0. b) M'(\u20133 ; 0 ; \u20132) ; 8. a) n = \u20134 , m = 4 ; c) MH = 2 3. b) n = \uf02d 10 , m = \uf02d9 \uf0d7 32 10. d(A, (A'BD)) = 1 ; d(A, (B'D'C)) = 2 \uf0d7 33 9. a) 5 ; b) 44 ; c) 2. 13 10. a) Ch\u00e4n h\u00d6 tr\u00f4c Oxyz sao cho A = (0 ; 0 ; 0), \u00a4n t\u00cbp ch\u2212\u00acng III B = (1 ; 0 ; 0), D = (0 ; 1 ; 0), A' = (0 ; 0 ; 1). Sau \u00ae\u00e3 vi\u00d5t ph\u2212\u00acng tr\u00d7nh c\u00f1a hai m\u00c6t 1. a) Vi\u00d5t ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (BCD), ph\u00bcng (AB'D') v\u03bc (BC'D), t\u00f5 \u00ae\u00e3 suy ra ch\u00f8ng minh A \uf0cf (BCD) ; ch\u00f3ng song song v\u00edi nhau. b) G\u00e3c gi\u00f7a hai \u00ae\u2212\u00eang th\u00bcng AB v\u03bc CD l\u03bc b) d = 3 \uf0d7 45o ; 3 c) AH = 1 (H l\u03bc ch\u00a9n \u00ae\u2212\u00eang vu\u00abng g\u00e3c h\u00b9 t\u00f5 A xu\u00e8ng (BCD)). 105","2. a) I(1 ; 1 ; 1) ; r = 62 ; 3. a) V \uf03d 1 \uf070 (2r \uf02d h)h2 ; b) (S) : (x \u2013 1)2 + (y \u2013 1)2 + (z \u2013 1)2 = 62 ; 3 c) (\uf061) : 5x + y \u2013 6z \u2013 62 = 0. b) V \u00ae\u00b9t gi\u00b8 tr\u00de l\u00edn nh\u00cat b\u00bbng 32\uf070 r3 3. a) (BCD) : 8x \u2013 3y \u2013 2z + 4 = 0 ; 81 b) AH = 36 ; 77 khi h = 4r . c) (\uf061) : x \u2013 z + 5 = 0. 3 \uf0ecx \uf03d 1\uf02b 2t \uf0ecx \uf03d 2 \uf02b 2t 4. a) d \/\/ AB ; b) \uf0ed\uf0efy \uf03d 3 \uf02d 4t b) I(2 ; 0 ; 4). \uf0ee\uf0efz \uf03d \uf02d5 \uf02d 5t . 5. a) V = 8 cm3 ; 4. a) \uf0ef\uf0edy \uf03d \uf02dt ; b) d(A, (BCD)) = 12 . 34 \uf0ef\uf0eez \uf03d \uf02d3 \uf02b 3t 5. T\u00a9m J(\u20131 ; 2 ; 3), b\u00b8n k\u00ddnh r' = 8. 6. a) S = 16 \uf070 a2 ; V = 32 \uf070 a3 ; 3 6. a) M(0 ; 0 ; \u20132) ; b) 4x + 3y + z + 2 = 0. b) O(0 ; 0 ; 0) ; r' = r = 2a ; 7. a) 6x \u2013 2y \u2013 3z + 1 = 0 ; c) Sxq = 4\uf070 a2 3 ; V = 4\uf070 a3 3 . b) M(1 ; \u20131 ; 3) ; \uf072\uf072 \uf0ecx \uf03d 1\uf02b 2t c) \uf0ed\uf0efy \uf03d \uf02d1 \uf02d 3t 7. a) \uf0ed\uf0ef\uf0ecad1 \uf0b9 kad2 ; \uf0ee\uf0efz \uf03d 3 \uf02b 6t. \uf0ee\uf0efd1 \uf0c7 d2 \uf03d \uf0c6 b) (\uf061) : 2x \u2013 y \u2013 3z \u2013 2 = 0. \uf075\uf075\uf072 \uf075\uf075\uf075\uf072 \uf075\uf075\uf075\uf072 8. a) Ch\u00f8ng minh AB, AC, AD kh\u00abng \u00ae\u00e5ng 8. 4x + 6y + 5z + 51 \uf0b1 5 77 = 0. ph\u00bcng. 9. H(\u20133 ; 1 ; \u20132). b) d(D, (ABC)) = 3 \uf02d 6 \uf02d 3 \uf03d 6 . 10. M'(6 ; 13 ; \u20134). 1\uf02b1\uf02b 4 \uf0ec 3 c) Ph\u2212\u00acng tr\u00d7nh c\u00f1a m\u00c6t c\u00c7u l\u03bc \uf0ef 7 \uf0ef x \uf03d (x \uf02d 3)2 + (y \uf02d 2)2 + (z \uf02d 0,5)2 = 41 . 4 11. \uf056: \uf0ed\uf0efy \uf03d \uf02d 25 \uf02b \uf0ef 7 t 1\uf0d7 1 \uf0d7 32 \uf0ef\uf0ee\uf0efz 18 d) VABCD \uf03d 21. 14. 6 \uf03d7. 7 \uf03d . 12. A'(\u20133 ; 2 ; 1). 9. a) V = 4 ; 3 \u00a4n t\u00cbp cu\u00e8i n\u00a8m 1. D\u00efng ph\u00d0p \u00ae\u00e8i x\u00f8ng t\u00a9m I. b) (S) : (x \uf02d 3)2 \uf02b (y \uf02d 3)2 \uf02b (z \uf02d1)2 \uf03d 21 ; 2. V \uf03d 25 a3 . 24 72 c) (\uf0611) : z \u2013 1 \u2013 21 = 0 ; 2 106 (\uf0612 ) : z \u2013 1 + 21 = 0. 2","10. a) A \uf0e6 \uf02d 10 ; 15 ; 5\uf0f6 ; \uf075\uf075\uf075\uf072 \uf075\uf075\uf072 \uf0e7\uf0e8 4 4 4 \uf0f7\uf0f8 14. a) G x\u00b8c \u00ae\u00denh b\u00ebi : AG \uf03d 2CB. b) (\uf062) : 4x \u2013 2y + 2z + 15 = 0. b) Ch\u00f8ng minh GM2 \uf03d k2 \uf02d (GA2 \uf02b 2GB2 \uf02d 2GC2 ). 11. a) (ABC) : 3x \u2013 5y \u2013 2z + 13 = 0 ; 15. a) (\uf061) : 2x \u2013 y \u2013 3z \u2013 2 = 0 ; \uf0ecx \uf03d \uf02d1\uf02b t (\uf062) : 2x \u2013 y \u2013 3z \u2013 1 = 0 ; AD : \uf0ed\uf0efy \uf03d 2 \uf02b t b) d(M' , (\uf061)) = d(M, (\uf062)) = 1 \uf0d7 \uf0ef\uf0eez \uf03d \uf02d2t. 14 b) (\uf061) : 5x \u2013 9y \u2013 2z + 23 = 0. \uf0ecx \uf03d t 12. a) (BCD) : x + 2y + 3z \u2013 7 = 0 ; 16. b) \uf0ef\uf0edy \uf03d 1 ; b) (S) : (x \u2013 3)2 + (y + 2)2 + (z + 2)2 = 14 ; c) H(4 ; 0 ; 1). \uf0ee\uf0efz \uf03d \uf02d1\uf02d 2t c) M'(\uf02d4 ; 0 ; \uf02d3) ; 13. a) d1 c\u00bet d2 t\u00b9i M(2 ; 3 ; 1). d) N'(\uf02d4 ; 0 ; 2). b) Ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng (d1, d2) l\u03bc 6x \u2013 8y + z + 11 = 0. 107","B\u00b6ng thu\u00cbt ng\u00f7 C 6 K 66 C\u00b9nh c\u00f1a h\u00d7nh \u00aea di\u00d6n Kho\u00b6ng c\u00b8ch gi\u00f7a hai \u00aei\u00d3m 48 Kho\u00b6ng c\u00b8ch t\u00f5 m\u00e9t \u00aei\u00d3m \u00ae\u00d5n m\u00e9t 78 D 33 m\u00c6t ph\u00bcng Di\u00d6n t\u00ddch m\u00c6t c\u00c7u 37 Kh\u00e8i b\u00b8t di\u00d6n \u00ae\u00d2u (h\u00d7nh t\u00b8m m\u00c6t 16 Di\u00d6n t\u00ddch to\u03bcn ph\u00c7n c\u00f1a h\u00d7nh n\u00e3n 32 \u00ae\u00d2u) 42 Di\u00d6n t\u00ddch to\u03bcn ph\u00c7n c\u00f1a h\u00d7nh tr\u00f4 36 Kh\u00e8i c\u00c7u Di\u00d6n t\u00ddch xung quanh c\u00f1a h\u00d7nh n\u00e3n Kh\u00e8i ch\u00e3p 4 Di\u00d6n t\u00ddch xung quanh c\u00f1a h\u00d7nh tr\u00f4 Kh\u00e8i \u00aea di\u00d6n 6 Kh\u00e8i \u00aea di\u00d6n \u00ae\u00d2u 15 \u00a7 Kh\u00e8i \u00aea di\u00d6n \u00ae\u00d2u lo\u00b9i {p ; q} 15 Kh\u00e8i \u00aea di\u00d6n l\u00e5i 14 \u00a7a di\u00d6n 6 Kh\u00e8i l\u00a8ng tr\u00f4 4 32 Kh\u00e8i l\u00cbp ph\u2212\u00acng 15 \u00a7\u00b8y c\u00f1a h\u00d7nh n\u00e3n 35 Kh\u00e8i l\u00cbp ph\u2212\u00acng \u00ae\u00acn v\u00de 21 5,6 Kh\u00e8i tr\u00f4 tr\u00dfn xoay (kh\u00e8i n\u00e3n) 32 \u00a7\u00b8y c\u00f1a h\u00d7nh tr\u00f4 46 Kh\u00e8i tr\u00f4 tr\u00dfn xoay (kh\u00e8i tr\u00f4) 36 5,6 Kh\u00e8i t\u00f8 di\u00d6n \u00ae\u00d2u 15 \u00a7i\u00d3m ngo\u03bci Kh\u00abng gian Oxyz 62 6 Kinh tuy\u00d5n 43 \u00a7i\u00d3m ti\u00d5p x\u00f3c 32 42 \u00a7i\u00d3m trong 31 35 \u00a7\u00d8nh c\u00f1a h\u00d7nh \u00aea di\u00d6n 46 45 \u00a7\u00d8nh c\u00f1a h\u00d7nh n\u00e3n \u00a7\u2212\u00eang k\u00ddnh m\u00c6t c\u00c7u \u00a7\u2212\u00eang sinh c\u00f1a m\u00c6t n\u00e3n \u00a7\u2212\u00eang sinh c\u00f1a m\u00c6t tr\u00f4 M \u00a7\u2212\u00eang th\u00bcng ti\u00d5p x\u00f3c v\u00edi m\u00c6t c\u00c7u M\u00c6t c\u00c7u 41 47 \u00a7\u2212\u00eang tr\u00dfn l\u00edn M\u00c6t c\u00c7u ngo\u00b9i ti\u00d5p h\u00d7nh \u00aea di\u00d6n 47 31 G M\u00c6t c\u00c7u n\u00e9i ti\u00d5p h\u00d7nh \u00aea di\u00d6n 45 G\u00e3c \u00eb \u00ae\u00d8nh c\u00f1a m\u00c6t n\u00e3n 44 31 M\u00c6t n\u00e3n tr\u00dfn xoay 62 31 M\u00c6t ph\u00bcng k\u00ddnh 35 32 H M\u00c6t ph\u00bcng ti\u00d5p x\u00f3c v\u00edi m\u00c6t c\u00c7u 35 Hai h\u00d7nh b\u00bbng nhau H\u00d6 to\u00b9 \u00ae\u00e9 Oxyz 10 M\u00c6t ph\u00bcng to\u00b9 \u00ae\u00e9 6 H\u00d6 tr\u00f4c to\u00b9 \u00ae\u00e9 \u00a7\u00d2-c\u00b8c vu\u00abng g\u00e3c 62 6 Oxyz M\u00c6t tr\u00dfn xoay H\u00d7nh \u00aea di\u00d6n 62 H\u00d7nh n\u00e3n tr\u00dfn xoay (h\u00d7nh n\u00e3n) 6 M\u00c6t tr\u00f4 tr\u00dfn xoay H\u00d7nh tr\u00f4 tr\u00dfn xoay (h\u00d7nh tr\u00f4) 32 M\u00c6t xung quanh c\u00f1a h\u00d7nh n\u00e3n 35 M\u00c6t xung quanh c\u00f1a h\u00d7nh tr\u00f4 Mi\u00d2n ngo\u03bci Mi\u00d2n trong 108","P 10 T 48 8 Th\u00d3 t\u00ddch kh\u00e8i c\u00c7u 23 Ph\u00a9n chia v\u03bc l\u00bep gh\u00d0p c\u00b8c kh\u00e8i 8 Th\u00d3 t\u00ddch kh\u00e8i ch\u00e3p 21 \u00aea di\u00d6n 9 Th\u00d3 t\u00ddch kh\u00e8i \u00aea di\u00d6n 22 Ph\u00d0p bi\u00d5n h\u00d7nh trong kh\u00abng gian 9 Th\u00d3 t\u00ddch kh\u00e8i h\u00e9p ch\u00f7 nh\u00cbt 23 Ph\u00d0p d\u00eai h\u00d7nh trong kh\u00abng gian 9 Th\u00d3 t\u00ddch kh\u00e8i l\u00a8ng tr\u00f4 34 Ph\u00d0p \u00ae\u00e8i x\u00f8ng qua \u00ae\u2212\u00eang th\u00bcng 8 Th\u00d3 t\u00ddch kh\u00e8i n\u00e3n 37 Ph\u00d0p \u00ae\u00e8i x\u00f8ng qua m\u00c6t ph\u00bcng Th\u00d3 t\u00ddch kh\u00e8i tr\u00f4 tr\u00dfn xoay 70 Ph\u00d0p \u00ae\u00e8i x\u00f8ng t\u00a9m 74 T\u00ddch c\u00e3 h\u2212\u00edng 65 Ph\u00d0p t\u00denh ti\u00d5n 66 T\u00ddch v\u00ab h\u2212\u00edng 44 Ph\u2212\u00acng tr\u00d7nh c\u00f1a m\u00c6t ph\u00bcng theo Ti\u00d5p di\u00d6n c\u00f1a m\u00c6t c\u00c7u 46 \u00aeo\u00b9n ch\u00ben 83 Ti\u00d5p tuy\u00d5n c\u00f1a m\u00c6t c\u00c7u 63 Ph\u2212\u00acng tr\u00d7nh m\u00c6t c\u00c7u To\u00b9 \u00ae\u00e9 c\u00f1a \u00aei\u00d3m 64 Ph\u2212\u00acng tr\u00d7nh tham s\u00e8 c\u00f1a 71 To\u00b9 \u00ae\u00e9 c\u00f1a vect\u00ac 31 \u00ae\u2212\u00eang th\u00bcng Tr\u00f4c c\u00f1a m\u00c6t n\u00e3n 31 Ph\u2212\u00acng tr\u00d7nh t\u00e6ng qu\u00b8t c\u00f1a Tr\u00f4c c\u00f1a m\u00c6t tr\u00dfn xoay 35 m\u00c6t ph\u00bcng Tr\u00f4c c\u00f1a m\u00c6t tr\u00f4 62 V 69 Vect\u00ac \u00ae\u00acn v\u00de 43 Vect\u00ac ph\u00b8p tuy\u00d5n V\u00dc tuy\u00d5n 109","M\u00f4c l\u00f4c Trang Ch\u2212\u00acng I. Kh\u00e8i \u00aea di\u00d6n 4 \u00a71. Kh\u00b8i ni\u00d6m v\u00d2 kh\u00e8i \u00aea di\u00d6n 4 I- Kh\u00e8i l\u00a8ng tr\u00f4 v\u03bc kh\u00e8i ch\u00e3p 5 II- Kh\u00b8i ni\u00d6m v\u00d2 h\u00d7nh \u00aea di\u00d6n v\u03bc kh\u00e8i \u00aea di\u00d6n 8 III- Hai \u00aea di\u00d6n b\u00bbng nhau 10 IV- Ph\u00a9n chia v\u03bc l\u00bep gh\u00d0p c\u00b8c kh\u00e8i \u00aea di\u00d6n 12 B\u03bci t\u00cbp 12 B\u03bci \u00ae\u00e4c th\u00aam. \u00a7\u00denh ngh\u00dca \u00aea di\u00d6n v\u03bc kh\u00e8i \u00aea di\u00d6n 14 \u00a72. Kh\u00e8i \u00aea di\u00d6n l\u00e5i v\u03bc kh\u00e8i \u00aea di\u00d6n \u00ae\u00d2u 14 I- Kh\u00e8i \u00aea di\u00d6n l\u00e5i 15 II- Kh\u00e8i \u00aea di\u00d6n \u00ae\u00d2u 18 B\u03bci t\u00cbp 19 B\u03bci \u00ae\u00e4c th\u00aam. H\u00d7nh \u00aea di\u00d6n \u00ae\u00d2u 21 \u00a73. Kh\u00b8i ni\u00d6m v\u00d2 th\u00d3 t\u00ddch c\u00f1a kh\u00e8i \u00aea di\u00d6n 21 I- Kh\u00b8i ni\u00d6m v\u00d2 th\u00d3 t\u00ddch kh\u00e8i \u00aea di\u00d6n 23 II- Th\u00d3 t\u00ddch kh\u00e8i l\u00a8ng tr\u00f4 23 III- Th\u00d3 t\u00ddch kh\u00e8i ch\u00e3p 25 B\u03bci t\u00cbp 26 \u00a4n t\u00cbp ch\u2212\u00acng I 27 C\u00a9u h\u00e1i tr\u00bec nghi\u00d6m ch\u2212\u00acng I Ch\u2212\u00acng II. M\u00c6t n\u00e3n, m\u00c6t tr\u00f4, m\u00c6t c\u00c7u 30 30 \u00a71. Kh\u00b8i ni\u00d6m v\u00d2 m\u00c6t tr\u00dfn xoay 31 I- S\u00f9 t\u00b9o th\u03bcnh m\u00c6t tr\u00dfn xoay 35 II- M\u00c6t n\u00e3n tr\u00dfn xoay 39 III- M\u00c6t tr\u00f4 tr\u00dfn xoay B\u03bci t\u00cbp 41 41 \u00a72. M\u00c6t c\u00c7u 43 I- M\u00c6t c\u00c7u v\u03bc c\u00b8c kh\u00b8i ni\u00d6m li\u00aan quan \u00ae\u00d5n m\u00c6t c\u00c7u 45 II- Giao c\u00f1a m\u00c6t c\u00c7u v\u03bc m\u00c6t ph\u00bcng III- Giao c\u00f1a m\u00c6t c\u00c7u v\u00edi \u00ae\u2212\u00eang th\u00bcng. Ti\u00d5p tuy\u00d5n c\u00f1a m\u00c6t c\u00c7u 110","IV- C\u00abng th\u00f8c t\u00ddnh di\u00d6n t\u00ddch m\u00c6t c\u00c7u v\u03bc th\u00d3 t\u00ddch kh\u00e8i c\u00c7u 48 B\u03bci t\u00cbp 49 \u00a4n t\u00cbp ch\u2212\u00acng II 50 C\u00a9u h\u00e1i tr\u00bec nghi\u00d6m ch\u2212\u00acng II 51 B\u00b9n c\u00e3 bi\u00d5t. Nh\u00f7ng v\u00can \u00ae\u00d2 li\u00aan quan \u00ae\u00d5n kinh tuy\u00d5n v\u03bc 55 v\u00dc tuy\u00d5n c\u00f1a Tr\u00b8i \u00a7\u00cat Ch\u2212\u00acng Iii. Ph\u2212\u00acng ph\u00b8p to\u00b9 \u00ae\u00e9 trong kh\u00abng gian 62 62 \u00a71. H\u00d6 to\u00b9 \u00ae\u00e9 trong kh\u00abng gian 64 I- To\u00b9 \u00ae\u00e9 c\u00f1a \u00aei\u00d3m v\u03bc c\u00f1a vect\u00ac 65 II- Bi\u00d3u th\u00f8c to\u00b9 \u00ae\u00e9 c\u00f1a c\u00b8c ph\u00d0p to\u00b8n vect\u00ac 66 III- T\u00ddch v\u00ab h\u2212\u00edng 68 IV- Ph\u2212\u00acng tr\u00d7nh m\u00c6t c\u00c7u B\u03bci t\u00cbp \u00a72. Ph\u2212\u00acng tr\u00d7nh m\u00c6t ph\u00bcng 69 I- Vect\u00ac ph\u00b8p tuy\u00d5n c\u00f1a m\u00c6t ph\u00bcng II- Ph\u2212\u00acng tr\u00d7nh t\u00e6ng qu\u00b8t c\u00f1a m\u00c6t ph\u00bcng 69 III- \u00a7i\u00d2u ki\u00d6n \u00ae\u00d3 hai m\u00c6t ph\u00bcng song song, vu\u00abng g\u00e3c 71 IV- Kho\u00b6ng c\u00b8ch t\u00f5 m\u00e9t \u00aei\u00d3m \u00ae\u00d5n m\u00e9t m\u00c6t ph\u00bcng 74 B\u03bci t\u00cbp 78 80 \u00a73. Ph\u2212\u00acng tr\u00d7nh \u00ae\u2212\u00eang th\u00bcng trong kh\u00abng gian 81 I- Ph\u2212\u00acng tr\u00d7nh tham s\u00e8 c\u00f1a \u00ae\u2212\u00eang th\u00bcng 82 II- \u00a7i\u00d2u ki\u00d6n \u00ae\u00d3 hai \u00ae\u2212\u00eang th\u00bcng song song, c\u00bet nhau, ch\u00d0o nhau 84 B\u03bci t\u00cbp 89 \u00a4n t\u00cbp ch\u2212\u00acng III 91 C\u00a9u h\u00e1i tr\u00bec nghi\u00d6m ch\u2212\u00acng III 94 B\u03bci \u00ae\u00e4c th\u00aam. Ch\u00efm m\u00c6t ph\u00bcng 97 \u00a4n t\u00cbp cu\u00e8i n\u00a8m 99 H\u2212\u00edng d\u00c9n gi\u00b6i b\u03bci t\u00cbp v\u03bc \u00ae\u00b8p s\u00e8 103 111","Ch\u00deu tr\u00b8ch nhi\u00d6m xu\u00cat b\u00b6n : Ch\u00f1 t\u00dech H\u00e9i \u00ae\u00e5ng Th\u03bcnh vi\u00aan NGUY\u00d4N \u00a7\u00f8c th\u00b8i T\u00e6ng Gi\u00b8m \u00ae\u00e8c ho\u03bcng l\u00aa b\u00b8ch Ch\u00deu tr\u00b8ch nhi\u00d6m n\u00e9i dung : T\u00e6ng bi\u00aan t\u00cbp PHAN XU\u00a2N TH\u03bcNH Bi\u00aan t\u00cbp l\u00c7n \u00ae\u00c7u : ho\u03bcng ng\u00e4c ph\u2212\u00acng \u2014 l\u00f4c v\u00a8n h\u03bco Bi\u00aan t\u00cbp t\u00b8i b\u00b6n : l\u00f4c v\u00a8n h\u03bco Tr\u00d7nh b\u03bcy b\u00d7a v\u03bc minh ho\u00b9 : NGuy\u00d4n m\u00b9nh h\u00efng S\u00f6a b\u00b6n in : PH\u00dfNG S\u00f6A B\u00b6N IN (NXBGD t\u00b9i TP. HCM) Ch\u00d5 b\u00b6n : PH\u00dfNG ch\u00d5 b\u00b6n (NXBGD t\u00b9i TP. HCM) H\u00d7nh h\u00e4c 12 112"]