Phương trình mặt phẳng

Home Lớp 12 Toán lớp 12 - Sách Kết nối tri thức Bài 14. Phương trình mặt phẳngBài tập nâng cao

{"common":{"save":0,"post_id":"8211","level":3,"total":10,"point":10,"point_extra":0},"segment":[{"id":"6048","post_id":"8211","mon_id":"1159285","chapter_id":"1159382","question":"Cho hai m\u1eb7t ph\u1eb3ng $(\\alpha):3x-2y+2z+7=0$<\/span>, $(\\beta):5x-4y+3z+1=0$<\/span>. Ph\u01b0\u01a1ng trình m\u1eb7t ph\u1eb3ng \u0111i qua g\u1ed1c t\u1ecda \u0111\u1ed9 O \u0111\u1ed3ng th\u1eddi vuông góc v\u1edbi c\u1ea3 (α) và (β) là:<\/p>","options":["A.<\/strong> $2x-y-2z-1=0$<\/span>","B.<\/strong> $2x-y+2z=0$<\/span>","C.<\/strong> $2x+y-2z=0$<\/span>","D.<\/strong> $2x+y-2z+1=0$<\/span>"],"correct":"3","level":"3","hint":"","answer":"Ch\u1ecdn C.<\/strong> $2x+y-2z=0$<\/span>.<\/span><\/p>Vector pháp tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng l\u1ea7n l\u01b0\u1ee3t là $\\overrightarrow{n}_\\alpha=(3;-2;2)$<\/span>, $\\overrightarrow{n}_\\beta=(5;-4;3)$<\/span>.<\/p>Khi \u0111ó $[\\overrightarrow{n}_\\alpha;\\overrightarrow{n}_\\beta]=(2;1;-2)$<\/span>.<\/p>Ph\u01b0\u01a1ng trình m\u1eb7t ph\u1eb3ng \u0111i qua g\u1ed1c t\u1ecda \u0111\u1ed9 O có vector pháp tuy\u1ebfn $\\overrightarrow{n}=(2;1;-2)$<\/span> là $2x+y-2z=0$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-25 09:06:08","option_type":"math","len":0},{"id":"6051","post_id":"8211","mon_id":"1159285","chapter_id":"1159382","question":"Trong không gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m A(2 ; 4 ; 1), B(–1 ; 1 ; 3) và m\u1eb7t ph\u1eb3ng $(P):x-3y+2z-5=0$<\/span>. M\u1ed9t m\u1eb7t ph\u1eb3ng (Q) \u0111i qua hai \u0111i\u1ec3m A, B và vuông góc v\u1edbi m\u1eb7t ph\u1eb3ng (P) có d\u1ea1ng $ax+by+cz-11=0$<\/span>. Kh\u1eb3ng \u0111\u1ecbnh nào sau \u0111ây là \u0111úng?<\/p>","options":["A.<\/strong> a + b + c = 5","B.<\/strong> a + b + c = 15","C.<\/strong> a + b + c = –15","D.<\/strong> a + b + c = –5"],"correct":"1","level":"3","hint":"","answer":"Ch\u1ecdn A.<\/strong> a + b + c = 5.<\/span><\/p>Vì (Q) vuông góc v\u1edbi (P) nên (Q) nh\u1eadn vector pháp tuy\u1ebfn $\\overrightarrow{n}=(1;-3;2)$<\/span> c\u1ee7a (P) làm m\u1ed9t vector ch\u1ec9 ph\u01b0\u01a1ng.<\/p>Vì (Q) \u0111i qua A và B nên (Q) nh\u1eadn $\\overrightarrow{AB}=(-3;-3;2)$<\/span> làm m\u1ed9t vector ch\u1ec9 ph\u01b0\u01a1ng.<\/p>Khi \u0111ó (Q) có m\u1ed9t vector pháp tuy\u1ebfn là $\\overrightarrow{n}_Q=[\\overrightarrow{AB};\\overrightarrow{n}]=(0;8;12)$<\/span>.<\/p>V\u1eady (Q): $0(x+1)+8(y-1)+12(z-3)=0$<\/span> hay (Q): $2y+3z-11=0$<\/span>.<\/p>Khi \u0111ó a = 0; b = 2 ; c = 3 nên a + b + c = 5.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-25 09:20:01","option_type":"txt","len":2},{"id":"6054","post_id":"8211","mon_id":"1159285","chapter_id":"1159382","question":"Trong không gian Oxyz cho hai m\u1eb7t ph\u1eb3ng $(P):x-3y+2z-1=0$<\/span> và $(Q):x-z+2=0$<\/span>. . M\u1eb7t ph\u1eb3ng (α) vuông góc v\u1edbi c\u1ea3 (P) và (Q) \u0111\u1ed3ng th\u1eddi c\u1eaft tr\u1ee5c Ox t\u1ea1i \u0111i\u1ec3m có hoành \u0111\u1ed9 b\u1eb1ng 3. Ph\u01b0\u01a1ng trình c\u1ee7a m\u1eb7t ph\u1eb3ng (α) là<\/p>","options":["A.<\/strong> x + y + z – 3 = 0","B.<\/strong> x + y + z + 3 = 0","C.<\/strong> –2x + y + z + 6 = 0","D.<\/strong> –2x + y + z – 6 = 0"],"correct":"1","level":"3","hint":"","answer":"Ch\u1ecdn A.<\/strong> x + y + z – 3 = 0.<\/span><\/p>(P) có vector pháp tuy\u1ebfn là $\\overrightarrow{n}_P=(1;-3;2)$<\/span>, (Q) có vector pháp tuy\u1ebfn là $\\overrightarrow{n}_Q=(1;0;-1)$<\/span>.<\/p>Vì (α) vuông góc v\u1edbi (P) và (Q) nên (α) có m\u1ed9t vector pháp tuy\u1ebfn là $\\overrightarrow{n}=[\\overrightarrow{n}_P,\\overrightarrow{n}_Q]=(3;3;3)$<\/span> hay $\\overrightarrow{n}_\\alpha=(1;1;1)$<\/span>  (1).<\/p>Vì (α) c\u1eaft tr\u1ee5c Ox t\u1ea1i \u0111i\u1ec3m có hoành \u0111\u1ed9 b\u1eb1ng 3 nên (α) \u0111i qua \u0111i\u1ec3m M(3 ; 0 ; 0)  (2).<\/p>T\u1eeb (1) và (2) ta suy ra (α): $1(x-3)+1(y-0)+1(z-0)=0$<\/span> hay (α): $x+y+z-3=0$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-25 09:29:18","option_type":"txt","len":2}]}