bài tập phương trình mặt phẳng trong hình không giản giải tích lớp 12 được soạn thảo theo phương pháp trác nghiệm và có hướng dẫn đáp án chi tiết phục vụ cho giáo viên và học sinh trong kì thi trung học phổ thông quốc gia
Bi PHNG TRèNH MT PHNG Cõu Trong khụng gian vi h to r ( b = 2;3; r a ( 1; 1; ) M ( 1;1;1) Oxyz , mt phng i qua im v nhn v ) lm cp vect ch phng, cú phng trỡnh l: x + y - z - 1= x - z - 1= x - z +1= A B C Hng dn gii: r a = ( 1; 1; ) uuur r r VTPT n( ) = a, b = ( 2;0; 1) r b = ( 2;3; ) qua M ( 1;1;1) ( ) : 2x z = r ( ) VTPT n = ( 2;0; 1) x - y + z - = D Chn A Oxyz Cõu Trong khụng gian vi h to , mt phng no cú phng trỡnh sau õy l mt phng i qua A ( 0; 1; ) , B ( 1; 2; ) , C ( 0;0; ) im ? x + y + z + = x + y + z + = A B x - y + z + = x + y - z + = C D Hng dn gii: uuu r AB = ( 1;3; ) uuur uuu r uuur VTPT n( ) = AB, AC = ( 7; 4;11) uuur AC = ( 0;1; ) qua A ( 0; 1; ) ( ) : 7x + y + z + = r ( ) VTPT n = ( 7; 4;11) Chn cõu A Oxyz Cõu () A ( 5; 2; ) , B ( 3; 4;1) Trong khụng gian vi h to , cho mt phng i qua hai im r a ( 1;1;1) () v cú mt vect ch phng l Phng trỡnh ca mt phng l: x + y - z - = x + y - 14 z - = A B x - y - z + = x + y + z + = C D Hng dn gii: uuu r AB = ( 8;6;1) uuur uuu r n( ) = AB, a = ( 5;9; 14 ) r a = ( 1;1;1) qua A ( 5; 2;0 ) ( ) : x + y 14 z = ( ) VTCP n( ) = ( 5;9; 14 ) Chn cõu B ( ) Oxyz Cõu Trong khụng gian vi h to , gi l mt phng ct ba trc ta ti ba im A ( 2; 0;0 ) , B ( 0; 3; ) , C ( 0;0; ) () Phng trỡnh ca mt phng l: (Chỳ ý: khụng cú cỏc ỏp ỏn) Hng dn gii: uuu r AB = ( 2; 3;0 ) uuur uuu r uuur VTPT n( ) = AB, AC = ( 6; 4; ) uuur AC = ( 2;0; ) qua A ( 2;0; ) ( ) : x + y 3z + 12 = r ( ) VTPT n = 6; 4; ( ) () Oxyz Cõu Trong khụng gian vi h to , gi A ( 5; 4;3) l mt phng qua cỏc hỡnh chiu ca lờn () cỏc trc ta Phng trỡnh ca mt phng 12 x +15 y + 20 z - 60 = A x y z + + = C Hng dn gii: A, B, C l: (dựng pt on chn) 12 x +15 y + 20 z + 60 = B x y z + + - 60 = D Ox, Oy, Oz A ' ( 5; 0; ) ; B ' ( 0; 4;0 ) ; C ' ( 0; 0;3 ) Gi ln lt l hỡnh chiu ca A lờn Ta cú: uuuur AB = ( 5; 4;0 ) uuur uuuur uuuur VTPT n( ) = AB, AC = ( 12;15; 20 ) uuuur AC = ( 5;0;3) qua A ( 5; 4;3) ( ) :12 x + 15 y + 20 z 60 = r VTPT n = ( 12;15; 20 ) ( ) Chn A A ( 2; 1;1) , B ( 1;0; ) , C ( 0; 2; 1) Oxyz Trong khụng gian vi h to , cho ba im trỡnh mt phng qua A v vuụng gúc vi ng thng BC l: x + y + 5z = Phng x + y 5z + = A B x + y + 5z = x y + 5z = C D Hng dn gii uuur BC = (1; 2; 5) Vộc-t ch phng (VTCP) ng thng BC l A ( 2; 1;1) Phng trỡnh mt phng i qua im v vuụng gúc vi ng thng BC nờn cú r uuur n = BC = (1; 2; 5) VTPT ( x 2) 2( y + 1) 5( z 1) = x + y + z = Phng trỡnh mt phng l: Oxyz Trong khụng gian vi h to , phng trỡnh mt phng trung trc ca on AB vi A ( 3; 1; ) , B ( 3;1;2 ) l: 3x y = A 3x + y = B x 3y = C x + 3y = D Hng dn gii ur uuur n = AB = ( 6; 2;0 ) VTPT ca mt phng l M ( 0;0; ) M AB Ta trung im l: 6( x 0) + 2( y 00 ) + 0( z 2) = x y = Phng trỡnh mt phng: A ( 3;1; 1) , Oxyz Trong khụng gian vi h to , phng trỡnh mt phng i qua hai im B ( 2; 1; ) v song song vi trc Ox l: yz =0 y + 2z = B A y + z = C D 3x + z = Hng dn gii uuu r AB = (1; 2; 5) r k = (1; 0;0) uu r uuu r r v nP = AB k = (0;5; 2) mp(P): ẹi qua A(3;1; 1) uur Coự VTPT n P = (0; 5; 2) 0( x 3) + 5( y 1) + 2( z + 1) = y + z = Phng trỡnh mt phng: Oxyz Trong khụng gian vi h to , phng trỡnh mt phng i qua hai im A ( 3;1; - 1) , B ( 2; - 1; 4) x - y +3z + = v vuụng gúc vi mt phng x - 13 y - z + = l: x - y - 5z +3 = A B 13 x - y - z + = x + y +5z - = D C Hng dn gii uuu r AB = (1; 2; 5) uuu r nmp = (2; 1;3) uur uuu r uuu r v nP = AB nmp = (1;13;5) mp(P): ẹi qua A(3;1; 1) uur Coự VTPT n = ( 1;13;5) P ( x 3) + 13( y 1) + 5( z + 1) = Phng trỡnh mt phng: x 13 y z + = () Oxyz Trong khụng gian vi h to , cho M ( 1;3; - 2) l mt phng i qua im v song x - y + 3z + = song vi mt phng x - y +3z + = A Phng trỡnh ca mt phng l: x - y + 3z = B x - y +3z - = x - y +3z + = D C Hng dn gii x - y +3z + D = Phng trỡnh mt phng (P) cú dng: M ( 1;3; - 2) mp(P) i qua : 2.1- + - + D = ị D = x - y + 3z + = Vy phng trỡnh mt phng (P): Trong khụng gian vi h to () Oxyz , gi A ( 2; - 1;5) l mt phng i qua im 3x - y + z + = vi hai mt phng cú phng trỡnh v vuụng gúc x - y + 3z +1 = v Phng trỡnh mt () phng l: x +2y + z - = 3x + y - = B A 3x - y - z + = C D 3x - z = Hng dn gii uur nP = ( 3; 2;1) uur nQ = ( 5; 4;3) Ta cú: v ur uur uur n = nP , nQ = ( 2; 4; ) ( ) VTPT ca mp l : ẹi qua A ( 2; 1;5) uur ( ) Coự VTPT n = ( 2; 4; ) mp : 2( x 2) 4( y + 1) 2( z 5) = x + y + z = Phng trỡnh mt phng: M ( 2; - 3;1) Oxyz Trong khụng gian vi h to song vi mt phng (Oyz) l: x- =0 A B , phng trỡnh mt phng i qua im x +2 = 2x + y = C v song x - y +1 = D Hng dn gii Phng trỡnh mt phng (P) song song vi mt phng (Oyz) cú dng: M ( 2; - 3;1) mp(P) i qua : + D = ị D =- Vy phng trỡnh mt phng (P): x- =0 Trong khụng gian vi h to , gi ( P) Oxyz x+D =0 M ( 0; 2;1) l mt phng i qua im v i qua giao ( ) : 3x y z + = ( ) : x + y + z 13 = tuyn ca hai mt phng: = v Phng trỡnh ( P) ca l: x+ y +z- 3=0 2x + y + z - = B A x- y +z- 3=0 2x - y + z +3 = C D Hng dn gii m ( x + y + z 13) + n ( 3x y 5z + 1) = Phng trỡnh chựm mt phng cú dng: M ( 0; 2;1) Phng trỡnh mp(P) i qua : m ( + 5.2 + 9.1 13) + n ( 3.0 5.1 + 1) = m n = Chn m =1 n =1 x+ y +z- 3=0 Phng trỡnh mp(P) l: M ( - 4;1; 2) Oxyz Trong khụng gian vi h to , mt phng i qua im phng trỡnh l: 2y- z =0 2y +z =0 2x - z = B C A v cha trc Ox cú y +z =0 D Hng dn gii M ( - 4;1; 2) mp(P) cha trc Ox v i qua im r i uuuu r OM r r mp(P) cha uuuu rgiỏ ca vect uur vr uuuu i = ( 1; 0;0 ) OM = ( 4;1; ) nP = i OM = ( 0; 2;1) , mp(P): ẹi qua M ( 4;1;2 ) r Coự VTPT n = (0; 2; 1) 0( x + 4) 2( y 1) + 1( z 2) = y z = Phng trỡnh mt phng: A( 2; - 1;6) , B ( - 3; - 1; - 4) , Oxyz 10 Trong khụng gian vi h to C ( 5; - 1;0) , cho t din ABCD vi D ( 1; 2;1) v Chiu cao ca t din ABCD k t nh A l: 3 A B uuur BC = ( 8;0; ) mp(P): C D Hng dn gii uuur uur uuur uuur BD = ( 4;3;5 ) nP = BC BD = ( 1; 2; ) , ẹi qua B ( 3; 1; ) , r Coự VTPT n P = (1; 2; 2) ( BCD) 1( x + 3) + 2( y + 1) 2( z + 4) = Phng trỡnh mt phng x + y z = : d A, ( BCD ) = 1.2 + 2.6 12 + 22 + (2) =5 Chiu cao ca t din ABCD k t nh A l: A( 2; - 1;1) , Oxyz 11 Trong khụng gian vi h to B ( - 2;1; - 1) , phng trỡnh mt phng i qua hai im 3x + y - z + = v vuụng gúc vi mt phng l: x - 5y - 7z = x - 5y - 7z +4 = x +5 y - z = B C A Hng dn gii uuu r uur uuu r uuu r nmp = (3; 2; 1) nP = AB nmp = (1; 5; 7) uuu r AB = (4; 2; 2) x +5 y + z = D v mp(P): ẹi qua A ( 2; 1;1) uur Coự VTPT n P = (1; 5; 7) 1( x 2) 5( y + 1) 7( z 1) = x y z = Phng trỡnh mt phng: 12 Trong khụng gian vi h to ( ) () Oxyz , cho hai mt phng v ( ) : x + ( m + 1) y + 3z = ( ) : ( n + 1) x y z = , m.n () Hai mt phng vi v ch tớch bng: - 10 B 10 A cú phng trỡnh: C D () v song song - Hng dn gii ( ) Ta cú: // A B C D = = ( ) A ' B ' C ' D ' m + m = = = n +1 n = m.n = 10 6 ( a ) : x + y + z +1 = Oxyz 13 Trong khụng gian vi h to , khong cỏch gia hai mt phng ( ) : x + y + 2z + = v A l: 2 B C D Hng dn gii ( ) Mt phng d [ ( ), ( ) ] = ( ) // nờn D D' 2 A + B +C 2 = +2 +2 2 ( a ) : x + y + z +1 = 0, Oxyz Cõu Trong khụng gian vi h to , cho ba mt phng ( b) : x + y - z + = 0, ( g) : x - y + = Trong cỏc mnh sau, mnh no sai? () ( ) () ( ) ( ) ( ) B C D () / /( ) A Hng dn gii r ( a) n1 (1;1; 2) A(1;0; 0) Mt phng cú VTPT l v cha im r ( b) n2 (1;1; 1) B (2; 0;0) Mt phng cú VTPT l v cha im r ( g) n3 (1; 1; 0) C(0;5;0) Mt phng cú VTPT l v cha im r r ( a) ( b) n1 (1;1; 2) n2 (1;1; 1) D thy, vect v khụng t l nờn khụng th song song vi Tớch vụ hng ca cỏc vect trờn u bng nờn ba mt phng trờn ụi mt vuụng gúc vi Chn ỏp ỏn A ( ) : x my + 3z + m + = Oxyz Cõu Trong khụng gian vi h to , cho hai mt phng ( ) : ( m + 3) x y + ( 5m + 1) z 10 = A Hng dn gii B ( a ) / / ( b) Hai mt phng Xột phng trỡnh Thay m =1 nu v () Vi giỏ tr no ca m thỡ C m m+6 = = m + 5m + 10 () v song song vi nhau? D (*) m = m = m + 3m = m = m + vo (*) ta cú: = = 10 m = Thay vo (*) ta cú: Chn ỏp ỏn A = 19 Vy Vy m =1 tha yờu cu bi toỏn m = khụng tha yờu cu bi toỏn A ( 5;1;3) , B ( 1; 6; 2) , C ( 5;0; 4) , D ( 4; 0; 6) Oxyz Cõu Trong khụng gian vi h to , cho bn im Mt () phng i qua hai im A, B v song song vi ng thng CD cú phng trỡnh l: 10 x + y + z - 74 = 10 x + y + z = A B 10 x - y + z + 74 = x +10 y - z - 74 = C D Hng dn gii uuu r uuur () AB(4;5; 1) CD (1; 0; 2) Ta cú v l hai vect cú giỏ song song vi mt phng nờn r uuur uuur n = AB, CD () l mt VTPT ca mt phng r uuu r uuur 1 4 n = AB, CD = ; ; ữ = (10;9;5) 2 1 Cú r A( 5;1;3) () n = (10;9;5) Do i qua v nhn l mt VTPT nờn cú phng trỡnh l: 10 x + y + z - 74 = Chn ỏp ỏn A ( ) Oxyz Cõu Trong khụng gian vi h to , mt phng M ( 5; 4;3) i qua im v ct cỏc tia Ox, A, B, C OA = OB = OC Oy, Oz ti cỏc im cho cú phng trỡnh l: x + y + z - 12 = x+y+z =0 A B x + y + z +3 = x- y +z =0 C D Hng dn gii () A, B, C OA = OB = OC Do mt phng ct cỏc tia Ox, Oy, Oz ti cỏc im cho nờn cú phng trỡnh dng: () x y z + + =1 a a a vi a>0 M ( 5; 4;3) Mt khỏc i qua im nờn ta cú: () x + y + z - 12 = Vy cú phng trỡnh l: + + =1 a = 12 a a a 10 A ( 0; 0; ) , Oxyz Cõu 52 Trong khụng gian vi h to , cho hỡnh lp phng ABCD.ABCD vi B ( 1;0;0 ) , D ( 0;1;0 ) , A ' ( 0; 0;1) gi M, N ln lt l trung im cỏc cnh AB v CD Tớnh khong cỏch gia hai ng thng AC v MN Mt hc sinh gii nh sau: uuuur uuuu r uuuur uuuu r A ' C ( 1;1; 1) , MN ( 0;1;0 ) A ' C , MN = ( 1;0;1) Bc 1: Ta cú: A ' ( 0; 0;1) () Bc 2: Mt phng cha AC v song song vi MN l mt phng qua v cú 1VTPT r n ( 1; 0;1) ( ) : x + z = d ( A ' C , MN ) = d ( M , ( ) ) + 1 = = 12 + 02 + 12 2 Bc 3: Ta cú: Bi gii trờn ỳng hay sai? Nu sai thỡ sai bc no? A ỳng B Sai bc1 C Sai bc D Sai bc Hng dn gii : C ( 1;1; ) , M ; 0;0 ữ, N ;1; ữ Ta cú ta uuuur uuuu r uuuur uuuu r A ' C ( 1;1; 1) , MN ( 0;1;0 ) A ' C , MN = ( 1;0;1) Ta cú: A ' ( 0;0;1) ( ) AC MN Mt phng cha v song song vi l mt phng qua v cú mt vect r n ( 1; 0;1) phỏp tuyn ( ) : x + z = Phng trỡnh mt phng + 1 d ( A ' C , MN ) = d ( M , ( ) ) = = 12 + 02 + 12 2 Ta cú: Bi gii cỏc bc ỳng Chn ỏp ỏn A 32 () Cõu 53 Mt phng M ( - 1; 2;3) i qua im v cha ng thng ỡù x = - 6t ùù ( d ) : y = 1- 4t ùù ùùợ z =- +15t Phng trỡnh ( ) mt phng l: 3x + y + z - = A x + y + 2z - = C 3x - y + z + = B x +3 y + z +9 = D Hng dn gii : uuuu r N ( 4;1; ) ( d ) => MN = ( 5; 1; ) Ta cú r d : u ( ) d =( - 6; - 4;15) Vec t ch phng ca ng thng uuuu r uu r ộMN , u ự= - 39; - 39; - 26 =- 13(3;3; 2) ) dỳ ỷ ( () r n = ( 3;3; ) Vy mt vect phỏp tuyn ca mt phng l ( ) 3( x + 1) + 3( y 2) + 2( z 3) = 3x + y + z + = => Phng trỡnh mt phng l Chn ỏp ỏn A () Oxyz Cõu 54 Trong khụng gian vi h to , cho mt phng i qua im v song song r r a ( 1; 2;3) b ( 3;0;5 ) () vi giỏ ca hai vect v Phng trỡnh ca mt phng l: 5x y + 3z + = x + y + z + = A B 10 x y z + 21 = x y 3z + 21 = C D Hng dn gii : r r r n = a ( ) , b = ( 10; 4;6 ) = ( 5; 2;3 ) Ta cú vec t ch phng ca mt phng l ( ) x + y + ( z + 1) = x + y + 3z + = Vy phng trỡnh mt phng l: Chn ỏp ỏn B A ( 0; 2;1) , B ( 3; 0;1) , C ( 1;0;0 ) Oxyz Cõu 55 M ( 0;0; 1) Trong khụng gian vi h to mt phng (ABC) l: , cho im Phng trỡnh 33 2x + 3y + z = A 2x + 3y 4z = x + y 8z + = C B 2x 3y 4z +1 = D Hng dn gii : uuur uuur uuu r uuur AB = ( 3; 2;0 ) , AC = ( 1; 2; 1) => AB, AC = ( 2;3; ) Ta cú r uuu r uuur n = AB, AC = ( 2;3; ) ( ABC ) Vy vec t phỏp tuyn ca mt phng l C ( 1; 0;0 ) ( ABC ) 2( x 1) + y z = x + y z = Phng trỡnh mt phng qua l: Chn ỏp ỏn B () Oxyz, Cõu 56 Trong khụng gian vi h trc to gi l mt phng ct trc to ti im M ( 8; 0; ) , N ( 0; 2; ) , P ( 0;0; ) A x y z + + =0 x y + 2z = C ( ) Mt phng ct () Phng trỡnh ca mt phng l: x y z + + =1 B x y + 2z = D Hng dn gii M ( 8;0;0 ) , N ( 0; 2;0 ) , P ( 0; 0; ) Ox, Oy, Oz ln lt ti : x y z + + =1 x y + 2z = Chn ỏp ỏn : D ( ) : x + y + z + = 0, Oxyz, Cõu 57 Trong khụng gian vi h trc to cho ba mt phng ( ) : x + y z + = 0, ( ) : x y + = () ( ) A uur n = ( 1,1, ) Ta cú : uur n = ( 1,1, 1) Trong cỏc mnh sau õy, mnh no sai ? ( ) ( ) ( ) // ( ) ( ) ( ) B C D Hng dn gii () l mt VTPT ca mt phng () l mt VTPT ca mt phng 34 uu r n = ( 1, 1, ) ( ) l mt VTPT ca mt phng uur uur n n = + = ( ) ( ) Do : uu r uur n n = + = ( ) ( ) uur uu r n n = + = ( ) ( ) Chn ỏp ỏn : C A ( 1;1;3 ) , B ( 1;3; ) , C ( 1; 2;3) Oxyz, Cõu 58 Trong khụng gian vi h to cho im Mt phng ( ABC ) cú phng trỡnh l: x + y + 2z = A x + y + 2z = C Gi r n x y + 3z = B x + y + 2z + = D Hng dn gii ( ABC ) l mt VTPT ca mt phng r uuu r uuur n = AB; AC = ( 1, 2, ) ( ABC ) : x + y + z = Ta cú Chn ỏp ỏn : C Cỏch khỏc: Thay to im A vo phng trỡnh ca cỏc phng ỏn ta tm chn phng ỏn B v C B Thay tip to im vo phng trỡnh phng ỏn B v C ta thu thy phng ỏn C tho Chn ỏp ỏn : C A ( 1;0; ) , B ( 0; 2; ) , C ( 0; 0;3) Oxyz Cõu 59 Trong khụng gian vi h to , cho Phng trỡnh no sau ( ABC ) õy khụng phi l phng trỡnh mt phng ? x y z + + =1 6x + 3y + 2z = A B 6x + 3y + 2z + = 12 x + y + z 12 = C D Hng dn gii 35 ( ABC ) Mt phng A ( 1; 0; ) , B ( 0; 2; ) , C ( 0; 0;3) Ox, Oy , Oz ct ln lt ti x y z ( ABC ) : + + = x + y + z = : Chn ỏp ỏn : C A ( 1;3; ) , B ( 1; 2; ) Oxyz, Cõu 60 Trong khụng gian vi h to cho hai im Phng trỡnh mt AB phng trung trc ca on thng l: x + y 12 z 17 = x + y + 12 z 17 = A B x y 12 z 17 = x y + 12 z + 17 = C D Hng dn gii () AB Gi l mt phng trung trc ca on I 0, , 1ữ I AB Gi l trung im ca on thng thỡ uuu r I 0, , ữ AB ( 2, 1, ) ( ) Mt phng i qua trung im v nhn lm mt VTPT : ( ) : x + y 12 z 17 = Chn ỏp ỏn : A A ( a;0;0 ) , B ( 0; b; ) , C ( 0; 0; c ) Oxyz Cõu 61 Trong khụng gian vi h to dng thay i cho 1 + + =2 a b c , cho a , b, c vi l nhng s ( ABC ) Mt phng luụn i qua im c nh l: 1 1 ; ; ữ ; ; ữ 1;1;1 2; 2; ( ) ( ) 2 2 A B C D Hng dn gii x y z ( ABC ) : + + = ( ABC ) Ox, Oy , Oz a b c Mt phng ct cỏc trc 2x y 2z 2x y 2z 1 + + =2 + + = + + a b c a b c a b c (*) 36 a , b, c > Do (*) ỳng vi mi Chn ỏp ỏn : C nờn ta ng nht cỏc t s 1 x = ,y = ,z = 2 2x y 2z 1 + + = + + a b c a b c Cn lu ý thờm vi HS: ch ng thc ỳng vi mi a,b,c Lỳc ú, cỏc h s tng ng v mi ng nht A ( 1; 2;1) Cõu 62 Trong khụng gian vi h to Oxyz, cho im v hai mt phng ( P ) : x + y z = ( Q ) : x + y 3z = , Mnh no sau õy ỳng ? ( Q) ( P) A Mt phng i qua im A v song song vi mt phng ( Q) ( P) B Mt phng khụng i qua im A v song song vi mt phng ( Q) ( P) C Mt phng i qua im A v khụng song song vi mt phng ( Q) ( P) D Mt phng khụng i qua im A v khụng song song vi mt phng Hng dn gii uur nP = ( 2, 4, ) ( P) Ta cú : l mt VTPT ca mt phng uur nQ = ( 1, 2, ) ( Q) l mt VTPT ca mt phng uur uur nP = 2nQ ( P ) ( Q) ( Q) song song hoc trựng + 2.2 3.1 = A ( Q ) M Chn ỏp ỏn : A Cỏch khỏc: = = ( P ) // ( Q ) Quan sỏt h s ta cú : + 2.2 3.1 = A ( Q ) M Chn ỏp ỏn : A 37 A ( 1; 2; ) Cõu 63 Trong khụng gian vi h to Oxyz, cho im , gi M, N, P ln lt l hỡnh chiu ( MNP ) vuụng gúc ca A lờn ba trc Ox, Oy, Oz phng trỡnh mt phng l: y z y z y z y z x + =1 x + + =1 x+ =0 x + +1 = 5 5 A B C D Hng dn gii A ( 1; 2; ) M ( 1;0;0 ) N ( 0; 2; ) P ( 0; 0; ) Ox, Oy , Oz im chiu lờn cỏc trc ln lt l , , x y z ( MNP ) : + = 1 Phng trỡnh mt phng Chn ỏp ỏn : A ( P) Oxyz, Cõu 64 Trong khụng gian vi h to A, B, C cho tam giỏc x + y z = A x + 3y 2z +1 = C ABC cho mt phng Ox, Oy, Oz ct ba trc G ( 1; 3; ) ln ltti ( P) cú trng tõm l Phng trỡnh mt phng x y z = B x + y z + 18 = D Hng dn gii x y z P : + + =1 ( ) A ( a; 0; ) , B ( 0; b; ) , C ( 0;0; c ) a b c Gi s Phng trỡnh a +0+0 = 0+b+0 ABC = a = 3 + + c b = c = = G ( 1; 3; ) l trng tõm tam giỏc x y z ( P) : + + = x + y z + 18 = Phng trỡnh hay Chn ỏp ỏn : D Oxyz, Cõu 65 Trong khụng gian vi h to cho hai im A, B Oy v song song vi trc cú phng trỡnh l: A ( 1; 1;5) , B ( 0; 0;1) l: ( P) Mt phng cha 38 A 4x z +1 = 4x + y z + = B y + 4z = D Hng dn gii uuu r AB ( 1;1; ) Ta cú r j ( 0;1;0 ) C 2x + z = , Oy l vộc t n v ca trc uuu r r AB, j = ( 4;0; 1) ( P) l mt vộc t phỏp tuyn ca mt phng ( P) : 4x z +1 = Chn ỏp ỏn : A ( P) Oxyz, Cõu 66 Trong khụng gian vi h to mt phng cha trc A ( 2; 3;5 ) Oz v im Mt ( P) phng cú phng trỡnh l: 2x + y = 3x + y = A B r k ( 0;0;1) 2x 3y = C Hng dn gii 3x y + z = D B ( 0;0;1) Oz Oz l vộc t n v ca trc v uuu r uuu r r AB = ( 2;3; ) AB; k = ( 3; 2;0 ) ( P) l mt vộc t phỏp tuyn ca ( P ) : 3x + y = Chn ỏp ỏn : B ( P) : x y = Oxy, Cõu 67 Trong khụng gian vi h to chiu vuụng gúc ca gc to bng: 600 450 A B cho mt phng O v l hỡnh ( Q) trờn mt phng uuur OH ( 2; 1; ) uur nP ( 1; 1;0 ) H ( 2; 1; ) 300 ( P) Gúc gia hai mt phng C Hng dn gii uuur OH =3 Q ( ) l mt vộc t phỏp tuyn ca ( P) D 900 ( Q) v r nP = l mt vộc t phỏp tuyn ca 39 Gi ( P) l gúc gia hai mt phng ( Q) v uuur r OH nP cos = uuur r = = 450 OH nP Chn ỏp ỏn : B d: Oxyz, Cõu 68 Trong khụng gian vi h trc to A ( 1; 2;3) Phng trỡnh mt phng 23 x 17 y z 14 = A 23 x 17 y z + 14 = C cho ng thng x y z + = = v im ( A, d ) l: 23x + 17 y + z 60 = B 23 x + 17 y z + 14 = D Hng dn gii r B ( 0;1; 3) u ( 3; 4;1) d ng thng i qua , nhn lm mt vộc t ch phng uuu r uuur r AB ( 1; 1; ) AB; u = ( 23; 17; 1) ( P) l mt vộc t phỏp tuyn ca ( P ) : 23 x 17 y z + 14 = Chn ỏp ỏn : C 40 41 42 43 44 45 46