Cẩm nang hướng dẫn luyện thi Đại học - Hình học: Phần 2

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Nối tiếp nội dung phần 1 tài liệu Cẩm nang luyện thi Đại học - Hình học, phần 2 giới thiệu các kiến thức chương 3 - Phương pháp tọa độ trong không gian bao gồm: Tích có hướng của hai vectơ và ứng dụng, lập phương trình mặt phẳng, phương trình đường thẳng,... Mời các bạn cùng tham khảo.

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Nội dung Text: Cẩm nang hướng dẫn luyện thi Đại học - Hình học: Phần 2

Cam nang luy?n tht VH tltnh hqc - Nguyen Tat Thu «-ty it\Tin miv UVVH Kharig Vi(t ^ . 2 9 _ 1_ Tam giac ABC la tam giac deu nen ABCD la t u di?n deu khi va chi khi Giaihe taduac H 15'15' 3 P = BD = CD = AB = 3V2. Ta c6 h? phuong trinh 3) Gpi I(x;y;z) la tam duong tron ngoai tiep tam giac ABC. Ta c6 ' ( x - 5 ) 2 + ( y - 3 ) 2 + ( z + l ) 2 = ( x - 2 ) 2 + ( y - 3 ) 2 + ( z + 4)2 A I ( x - 2 ; y - 3 ; z - l ) , B I ( x + l ; y - 2 ; z ) , a ( x - l ; y - l ; z + 2). (x-5)2+(y-3)2+(z + l)2=(x-l)2+(y-2)2+z2 ' • Vi I la tam duong tron ngo^i tiep tam giac ABC nen ( x - 5 ) 2 + ( y - 3 ) 2 + ( z + l)2=18 A I = BI AI^ = BI^ 6x + 2y + 2z = 9 lz = l - x z= l-x AT = CI
Liim nang luyfn ini VH ninn nyt. - jTjjwyt.). x„m - C t y TNHH MTV DWH Khang Vift Bai 3.1.7. Trong khong gian O x y z , cho hinh hop chu nhat A B C D . A ' B ' C ' D ' g^i 3.1.8. Trong khong gian voi h? tryc tpa dp Oxyz cho hinh chop S.ABCD A = 0 , B e 0 x , D e C ) y , A ' 6 0 z va AB = 1, A D = 2, A A ' = 3. CO day ABCD la hinh thang vuong tai A, B voi AB = BC = a; A D = 2a; A s O, 1) Tim tpa do cac dinh ciia hinh hpp. B thupc tia Ox, D thupc tia Oy va S thupc tia Oz. Duong t h i n g SC va 2) Tim diem E tren duong thSng D D ' sao cho B'E 1 A ' C R D tao v o i nhau mot goc a thoa cosa = — ^ . 3) Tim diem M thupc A ' C , N thupc BD sao cho M N 1 B D , M N 1 A ' C . V30 do tinh khoang each giua hai duang th^ng cheo nhau A ' C va BD . J) Xac dinh tpa dp cac dinh ciia hinh chop Huong dan giai 2) Chung minh rang ASCD vuong, tinh di?n tich tam giac SCD va tinh c6 sin 1) Taco A(0;0;0),B(1;0;0), D(0;2;0), A'(0;0;3). ciia goc hop boi hai mat phMng (SAB) va (SCD). H i n h chieu ciia C len (Oxy) la C, hinh chie'u ciia C len Oz la A nen 3) Gpi E la trung diem canh A D . Tim tpa dp tam va tinh ban kinh m|it cau ngoai tie'p hinh chop S.BCE . C(l;2;0). 4) Tren cac canh SA,SB,BC,CD Ian lupt lay cac diem M , N , P , Q thoa Hinh chieu ciia B',C',D' len mp(Oxy)va true Oz Ian lug^ la cac diein SM = M A , SN = 2NB, BP = 3PC, CQ = 4QD . Chung minh rang M , N , P, Q B,C,D va A ' nen B'(l;0;3), C'(l;2;3), D'(0;2;3). khong dong p h i n g va tinh the tich khoi chop M N P Q . 2) V i E thuoc duong thSng DD" nen E(0;2;Z) , suy ra B'E = ( - l ; 2 ; z - 3 ) ^^k.. Huong dan giai Ma A ^ = ( l ; 2 ; - 3 ) nen B'E 1 A ' C o B ^ . A ^ = 0 . o - l + 4 - 3 ( z - 3 ) = 0 o z = 4. Vay E ( 0 ; 2 ; 4 ) . 3) Dat A ' M = x.A'C; BN = y.BD Ta CO A M - A A ' + A ' M = A A ' + X.A'C = (x; 2x;3 - 3x), suy ra M (x;2x; 3 - 3x) A N = A B + B N = A B + y . B D = ( l - y;2y;0) N ( 1 - y;2y;0) MN.A'C = 0 Theo gia thiet cua debai, ta c6: MN.BD = 0 Ma M N = ( l - x - y ; 2 y - 2 x ; 3 x - 3 ) , A ' C = ( l ; 2 ; - 3 ) , BD = (-1;2;0) l _ x - y + 4y-4x-9x + 9 = 0 [-14x + 3y = -10 Nen (*) 44 H + x + y + 4y-4x = 0 -3x + 5y = l (17 88 ' l)Ta (53 106 24 CO A(0;0;0),B(a;0;0),D(0;2a;0),C(a;a;0).Dat SA = x S(0;0;x) .^.^ Do do M 6 1 ' 61 61 6?'61'" = (-a; 2a; O), SC = (a; a; - x ) ^ DB = a^/5, SC = > / ? T 2 ^ B D . S C = a^ Vi M N la duong vuong goc chung cua hai duong t h i n g A ' C , B D nen SCBD cos a = C O S ( S C , B D ) d ( A • C, BD) = M N = ^(1 - X - y ) ' + (2y - 2xf + (3x " 3)' = ^ • ^ + 2 a ^ = 6 a ^ c ^ x = 2a=^S(0;0;2a).
2) Taco CS = {-a;-a;2a),CD = (-a;a;0)=>CS.CD = 0=>ASCD vuong tai C. 'pai Trong khong gian vdi h? tpa dp Oxyz cho hinh hpp chu n h l t ABCD.A'B'C'D' c6 A triing voi goc tpa dp, B(a;0;0),D(0;a;0) ,A'(0;0;b) Do do: S^cD = ^CS.CD = i aV6.aV2 = 3 ^ 7 3 v6i (a > 0, b > 0) . Gpi M la trung diem cua C C . Goi P = ((SCD), (SAB)). Ta c6 hinh chieu cua tarn giac SCD len mat phang 1) Tinh the tich ciia khoi t u d i f n B D A ' M . 2) Cho a + b = 4 . Tim max V^.guj^ . c ^a.2a 1 Huong dan giai (SAB) la tam giac SAB nen ta suy ra cosP = ^ASAB _ 2 ^ASCD a^ \/3 \[3 1) Ta c6: C(a;a;0), B'(a;0;b), C'(a;a;b), D'(0;a;b) : ^ M ( a ; a ; | ) 3) Ta CO E(0;a;0). Gpi l ( x ; y ; z ) la tam mat cau ngoai tiep hinh chop SBCE Suyra A ^ = (a;0;-b), A ^ = (0;a;-b), A ^ = b IB^ =IS^ (x-a)2+y2 + z2=x2+y2+(z-2a)2 Khi do IC^ =IS^^< ( x - a ) 2 + ( y - a ) 2 + z 2 = x 2 + y 2 ^{z-2af a;a;-- lE^ =IS^ x^ + (y - af + z^ = x^ + y^ + (z - 2af nen A ' B , A ' D = (ab;ab;a2) = > A ' M . A ' B , A ' D 3a^h a a'b X= — Vay VA'MBD = -2x + 4z = 3a 2 a a 2) Do a, b > 0 nen ap dung BDT Co si ta c6: - x - y + 2z = a o ^ 2 i^2^^ -2y + 4z = 3a z=a 4 = a + b = l a + l a + b>33/ia2b^a2b N 3—• , BP = - B C ^ P a;—;0 Vay max v.. A'BDM 3 4 I 4 J 27 Bai 3.1.10. Trong khong gian Oxyz cho t u di?n deu ABCD c6 A ( 3 ; - l ; 2 ) va CQ = - C D = > Q 7^ 5 G ( l ; l ; l ) la trpng tam tam giac ABC. Duong thSng BC di qua M 4;4;-j 3a a 9a Suy ra M N = , MP = a ; - ; - a Xac djnh toa dp cac dinh con lai va h'nh the tich khoi t u d i f n do. ' 3' ' 3 Huong dan giai ^ 2 2 2^ M N A M P = a a a (MNAMP).MQ = — Taco: A G = ( - 2 ; 2 ; - l ) , A M - ( l ; 5 ; y ) 4 3 2 nen M , N , P, Q khong dong phang. S u y ra riABC = A G A 2 A M = (-12; - 2 4 ; -24) Phuong trinh (ABC) : x + 2 y + 2 z - 5 = 0. Vay V^,NPQ=^(MNAMP).MQ a 40' I Ug^ A G A HABC = ( 6 ; 3 ; - 6 )
- t y 1 i\tltl MIV DWH KhangVift § 2. LAP PHLTONG TRlNH M A T PHANG x = 4 + 2t De lap phuong trinh mat p h i n g (a), ta c6 cac each sau: Phuongtrinh B C : y =4 + t Cach 1: Tim mpt diem M(xo;yo;zo) ma mat p h a n g (a) d i qua va mpt VTPT n = (a;b;c). Khi do phuong trinh cua (a) c6 d a n g : Goi a la canh cua t u dien di?n A B C D , ta c6 A G = — = 3 =^ a = 3N/3 a ( x - X o ) + b ( y - y o ) + c ( z - Z o ) = 0. '^P-' ' Mpt so l u u y khi t i m VTPT cua mat phSng ( a ) : AB = 2t + l;t + 5 ; - 2 t - Y Ta c6: B 4 + 2t;4 + t ; - - - 2 t , Neu hai vec to a,b khong ciing phuong va c6 gia song song hoac n i m tren 11,2 (a) thi a A b = n la VTPT cua (a). Nen ta dugc phuong trinh : ( 2 t + lf+(t + 5)2+(2t + Y) - 2 7 , Neu mat ph5ng (a) di qua ba diem phan b i f t k h o n g thang hang A,B,C thi 73 AB A AC = n la VTPT cua (a). Ot2+4t + i ^= 0 o t = -2±- 4 Neu (a)//(p) thi n^ = np . f ^ . 4 + ^/3.1-2^/3 4-N/3 1 + 2N/3' , Neu A 1 (a) thi n^^ = u ^ . Suy ra B , C 2 2 . Neu (a)//(P) thi i^//(a). x - l +t . Neu A(a;0;0),B(0;b;0),C(0;0;c) voi abc^O t h i p h u o n g trinh (ABC): Do D G 1 (ABC) nen phuong trinh G D : \ y = 1 + 2t Suy ra D ( l + t ; l + 2 t ; l + 2t) z = l + 2t m Cach 2: Gia su phuong trinh (a) c6 dang: , Ma GD = N / D A ^ ^ ^ A G ^ = ^ = 3N/2 =^ 3 t =3^f2^i = ±^f2 ax + by + cz + d = 0 voi a^ + b^ + c^ > 0. 3 D o d o S(1 + N/2;1 + 2>/2;1 + 2V2) hoac S(l - N/2;1 - 2 ^ 2 ; ! - 2^/2). Dua vao gia thiet cua de bai ta tim dupe ba t r o n g bo'n an a,b,c,d theo an con lai. ChSng han a = mb,c = nb,d = p b . Khi do p h u o n g trinh (a) la: The t.chcuatudi?n: V = 1DG.SABC =— (
Cty TNHH MTV DWH Khang Vi$t Clin lui/iii Ihi nil Ilnili UOL \^uyenTafT}iH Led gidi. Chii y: , Neu (P) chua (hoac song song) vol A B thi gia cua vec to A B se nam tren 1) Taco AB(-l;-2;l), AC = (-3;2;-2) AB A AC = (2;-5;-8) I (hoac song song) voi (P). Suy ra n„ = (2; -5;-8) . Do do, phuang trinh ciia ( a ) la: ^„ , Neu (P) -L (Q) thi VTPT cua mat phSng nay se c6 gia nam tren hoac song 2 ( x - l ) - 5 ( y - l ) - 8 ( z - l ) = 0 hay 2 x - 5 y - 8 z +11 = 0. song voi mat phSng kia. 2) Goi Aj, A j , A3 laa lugt la hinh chie'u ciia A len cac true Ox, Oy, Oz. Vi dM 3.2.3. Viet phuong trinh mat phang (a) biet: 1) (a) chua A(l;l;2), B(1;-1;1) va song song voi CD trong do C(0; 3; 0), Ta CO Aj(l;0;0), A2(0;l;0), A 3 ( 0 ; 0 ; ; ) . D(-2;0;-l). Vi (a) di qua Aj,A,,A3 nen phuang trinh c6 dang: 2) (a) chua A , B va vuong goc voi mat phSng (p):x + y + z + l = 0 x y z 3) (a) di qua C va vuong goc voi hai mat phMng (P) va (y): 2x - y + z - 3 = 0 . - + — + — = 1 hay x + y + z - l = 0. I l l ^ ^ 4) (a) di qua A, B va each deu hai diem M(3;-5;-1), N(-5;-1;7) . Dang 2. Lap phuang trinh mat phang (a)song song v&i mat Lai gidi. phang Cp; hoac (a) 1 A B . 1) Taco A B = (0;-2;-l), CD = (-2;-3;-l). Ta CO n^^ hoac AB la VTPT cua ( a ) . Vi (a) di qua A, B va song song voi CD nen n^ = A B A CD = (-1;2; -4). Chii dv Vi 3.2.2.((3): y: Neu Lapaxphuong + by + cztrinh + d mat phSng = 0 thi np -(a) biet: (a;b;c). Phuang trinh (a): x - 2y + 4z - 7 = 0. 1) (a) diqua M(2;3;1) vasong song voim^t phing (P):2x + 3 y - z + l = 0 2) Taco n p = ( l ; l ; l ) . 2) (a) la mat ph^ng trung true cua do^n A B vol A(1;4;2), B(-3;2;-1) . Vi (a) chua A , B va song song voi (p) nen n^ = ABAnp ==(-l;-l;2). Lai gidi. ^ Phuong trinh mat phSng ( a ) : x + y - 2z + 2 = 0 . 1) Taco r^ = (2;3;-l). 3) Taco n^^ = ( 2 ; - l ; l ) . ,. Vi (a)//(p) nen n ^ = njj = (2; 3;-l) .Phuang trinh mat phing (a) la: Vi mat phSng (a) vuong goc voi hai mat phSng (P) va (y) 2 ( x - 2 ) + 3 ( y - 3 ) - l ( z - l ) = 0 hay 2x + 3 y - z - 1 2 = 0 . nen n„ = np A n^ = (2;l;-3). Phuang trinh mat phSng ( a ) : 2x + y - 3z - 3 = 0 . 2) Vi (a) la mat ph^ng trung tryc cua doan AB nen AB = (-4;-2;-3) la VTPT Vi (a) each deu hai diem M, N nen ta c6 cac truang hg-p sau ciia ( a ) . T H 1: M N / / ( a ) . K h i d 6 n^ = A B A M N = (-12;8;-16). Phuong trinh mat phang (a) la: Phuong trinh ( a ) : 3x - 2y + 4z - 9 = 0 . 4 ( x - l ) + 2 ( y - 4 ) + 3 ( z - 2 ) = 0 hay 4x + 2y + 3 z - 1 8 = 0. T H 2: M N c3t (a) tai I, suy ra I la trung diem doan M N . Dang 3: Lap phifdng trinh mat phang (a) biet (a) song song vdi hai vect(< D o d o I(-1;-3;3). khong cung phi^dng a,b Khi do n„' = A1 A BI = (-6; 2; -4). f huong trinh ( a ) : 3x - y + 2z - 6 = 0 . Khi do n„ = a A b .
c r y i j V f j H MTVnWHKhans: Vu-I Dang 4. Viet phuang trinh mat phdng di qua m6t diem , a = - ^ b => b = -2a, ta chpn a = 1 => b = c = - 2 . vd lien quan den hhodng cdch hoac goc phuong trinh mat p h i n g (a): x - 2y - 2z + 6 = 0 . Phuong trinh di qua M(xQ;yo;zQ) c6 dang 3 2 , a = — b = > b = — a , ta chon a = 3 => b =-2,c =-6. 2 3 a(x-XQ) + b ( y - y g ) + c ( z - Z o ) = 0 vai + >0. phuang trinh mat phang (a): 3x - 2y - 6z + 1 4 = 0 . Dua vdo dieu ki$n khoang each hoac goc ta tim du
, a = - - b , t a c h Q n b = - 2 = > a = l , c = 2. Dang 6: Phuong trinh doan chdn 2 N e u mat phSng (a) d i qua A(a;0;0),B(0;b;0),C(0;0;c), a b c ^ O t h i p h u a n g Phuang trinh (a) la: x - 2 y + 2 z - 4 = 0 . t r i n h (a) c6 d a n g - + f + - = 1 . ' -, a b c I.' .... :i Dang 5: Viet phuong trinh mat phdng di qua giao tuyen ciia '^jid^ 3.2.6.. L a p p h u o n g t r i n h m a t phSng d i qua d i e m M ( 1 ; 9 ; 4 ) va cSt cac hai mdt phdng (a) t r y c tpa d p tai cac d i e m A , B,C (khac goic tpa dp) sao cho Cho hai mat p h a n g ( a J i a j X + b j y + C j Z + d i = 0 va (a2):a2X + b 2 y + C2Z + d2 = 0 . 1) M la t r u e t a m cua t a m giac ABC. ' : 2) K h o a n g each t u goc tpa d p O deh m | t phSng ( a ) la I a n nhat. + b , y + CiZ + di =0 [aiX Xet he p h u a n g t r m h \ u i 3) O A = OB = O C . I a 2 X + b j y + C2Z + d 2 = 0 4) 8 0 A = 120B + 16 = 3 7 0 C va x ^ > 0,Zc < 0. Cho z b o i hai gia t r j , giai h ^ t i m d u p e hai bp (x;y) t u a n g u n g . G p i A, B la hai L o i giai. d i e m c6 tpa d p la hai n g h i ^ m ciia h ^ . K h i d o (a) d i qua giao t u y e n ciia ( a j ] Gia s u m a t p h a n g (a) cat cac t r y c tpa d p tai cac d i e m khac goc tpa d p la va ( a 2 ) A , B 6 ( a ) . A(a;0;0),B(0;b;0),C(0;0;c) v o i a , b , c ^ O . V i d y 3.2.5. Cho ba mat phang: X V z (a,): X + y + z - 3 = 0; ( t t j ) : 2x + 3y + 4z - 1 = 0 . Phuang t r i n h m a t p h a n g (a) c6 d a n g — + — + — = 1. a b c C h u n g m i n h rang hai m a t ph§ng ( a , ) va ( t t j ) dt n h a u . Viet p h u o n g 1 9 4 Mat phang (a) d i qua d i e m M ( l ; 9 ; 4 ) nen - + — + - = 1 (1). t r i n h (P) d i qua A ( 1 ; 0 ; 1 ) va giao t u y e n ciia ( a j ) va ( a 2 ) . a b c 1) Ta c6: A M ( 1 - a; 9; 4), BC(0; - b; c), BM(1; 9 - b; 4), CA(a; 0; - c); Lai giai. Diem M la true t a m t a m giac A B C k h i va chi k h i Ta CO n7=(l;l;l), n j =(2;3;4) Ian l u p t la VTPT cua hai mat p h i i n g (Oi) va (cxj)- 1 9 4 , Me (a) - + - + - = 1 V I n j ?t kn2 nen hai mat phang (QJ) va ( a 2 ) cat n h a u . a b c AM.BC = 0 o ^ a = 98;b = 28;e = i ^ x + y + z-3::=0 9b = 4c 9 2 Xet h ^ p h u a n g t r i n h BM.CA = 0 a = 4c 2x + 3 y + 4 z - l = 0 ' x+ y = 4 x-7 Cho z = - 1 , ta C O he .M(7;-3;-l)e(ai)n(a2) Phuang t r i n h m a t p h i n g (a) can t i m la x + 9y + 4z - 98 = 0. 2x + 3y = 5 ' y = -3^ x+y = 5 -1 • Cho z = - 2 , ta C O h? ''^^^^N(6;-l;-2)e(ai)n(a2)- ^achl: Taco: d(0,(a)) = 2x + 3y = 9 VI (a) d i qua giao t u y e n cua h a i m§t ph4ng {a^) v a ( t t z ) n e n ( a ) d i qu^ a^ b^ • c2 Va^ h'2 c„2 M,N. .--'^ Bai toan t r o thanh, t i m gia trj nho nhat ciia T = 2 - + A + ^ v d i cac so t h u c Ta C O A M - (6; - 3 ; - 2 ) , A M = (5; - 1 ; - 2 ) ^ A M A A N = (4; 2; 2 ) . a^ b2 c- Suy ra n ^ = ( 2 ; l ; l ) . P h u o n g t r i n h mat p h ^ n g ( a ) la: 2x + y + z - 3 = 0 . ^ ' b , c ; ^ 0 t h o a m a n ! + - + ! = 1 (]).' a b c "^P d u n g b d t Bunhiacopski ta c6: ,
L t y iTWHH MTV DWH Khang Viet 8 2s 4 37 , Neu b > 0 = ^ c = - — a , b = — - l - , a > 2 n e n t u (1) ta c6 a=5 i i + - ^ - ^ = l « a 2 + 2 a - 3 5 = 0«> 2a - 4 2a a = -7 Nen suy ra T > ^ . Dau dang thuc xay ra khi ' >• • 40 1:1 = 9 : 1 = 4:1 ^ y j a > 2 nen a = 5=>b = 2;c = , phuang trinh m|t ph3ng can t i m la • ^ ^ o a = 9b = 4c = 98. (a): 8x + 20y - 37z - 40 = 0. 1 9 4 , a b c ^ , Meu b < 0 = > c = - ^ a , b = - l ^ ^ , a > 2 nentu ( l ) t a c 6 Phuang trinh mat phang (a) can tim la x + 9y + 4z - 98 = 0. Cach 2: Gpi H la hinh chieu cua O tren mat phang (a). 1, |I = , « , 29a - 35 . 0 « a = Vi mat phang (a) luon d i qua diem co'djnh M nen a 4-2a 2a 2 d(0,(a)) = O H < O M = >/98. Vi a > 2 nen khong c6 gia tri thoa man. Dau dang thuc xay ra khi H = M , khi do (a) la mat phang d i qua M va c6 V^y phuang trinh mat phSng (a): 8x + 20y - 37z - 40 = 0. vec to phap tuyen la O M ( l ; 9 ; 4 ) nen phuang trinh (a) la 2) Tim tpa dp diem thupc m|t p h i n g 1 .(x -1) + 9(y - 9) + 4.(z - 4) = 0 « . X + 9y + 4z - 98 = 0. . Diem M(xQ;yo;zo)€(a):ax + by + cz + d = O o a x o + byo + cZo + d = 0. 3) Vi OA = OB = OC nen a = b = c , do do xay ra bo'n truong h^p sau: H G (P) • Truong hop 1: a = b = c. Hinh chie'u H ciia M len mp(P) dupe xac djnh boi -j . _ . Vi dv 3.2.7. Trong khong gian Oxyz cho ba diem M H = t.n„ Tu (1) Tu (1) suy ra 1 suy ra 1++ -—+1 = 1 a = 14, nen phuang trinh (a) la: aa aa a A(3;3;3),B(1;2;4),C(0;3;2) vam|tphSng ( a ) : x + y + z - 3 = 0. x + y + z - 1 4 = 0. 1) Tim tpa dp hinh chie'u cua A len (a). . Truang hap 2: a = b = -c. T u (1) suy ra 1 + ^ - 1 = 1 « a = 6, nen phuong j ) J i m tpa dp diem M thupc (a) sao cho M A + MB nho n h a t j'ong hop 2: a = 1 Laigidi. trinh (a) la x + y - z - 6 = 0. - '"^9i H la hinh chieu ciia A len (a) . 1 9 4 • Truong hgp 3: a = - b = c. T u (1) suy ra + - = l o a = -4, nen phuang Si Si Si x = 3+t Phuang trinh A H : y = 3 + t = > H ( 3 + t;3 + t;3 + t ) . trinh (a) la x - y + z + 4 = 0. 1 9 4 z=3+t • Truong hp-p 4: a = - b = -c. T u (1) c6 = la = -12, nen phuang Si Si Si ^» H e ( a ) nentaco: 3 +1+ 3 +1 + 3 + 1 - 3 = 0 o t = - 2 H(1;1;1) . trinh (a) la x - y - z +12 = 0. ^' ^oi mSi diem K ( x ; y ; z ) ta d?t f(K) = x + y + z - 3 . Vay CO bon mgt phang thoa man la x + y + z - ; 4 = 0, va cac mat phang CO f(A) = 6, f(B) = 4 => A , B nSm cung phia so voi m^t ph5ng ( a ) . x + y - z - 6 = 0 , x - y + z + 4 = 0, x - y - z + 12 = U. 4) V i x^ > 0,Z(-- < 0 nen a > 0,c < 0, do do A ' la diem dol xung voi A qua ( a ) , suy ra A ' ( - 2 ; - 2 ; - 2 ) ; 8 0 A = 120B + 16 = 370C o 8a = 12 b +16 = -37c. do, voi mpi diem M e ( a ) , ta c6: M A + MB = M A ' + MB ^ A ' B .
Cam nang luucti Ihi nil UinJi hoc - Nguyen Tat ITtu Dau "=" xay ra khi M = A' B m (a). 4) Vi (a) each deu A,0 nen ta c6 cac truang h^xp sau. x = -2 + 3t THl:AO//(a) Vi A'B = (3;4;6), phuang trinh A'B: y = -2 + 4t => M(-2 + 3t; -2 + 4t; -2 + 6t) Ta c6: B C = (-3; 3; -4), OA = (1; 1; 1) =^ B C A OA = (7; -1; - 6 ) . z = -2 + 6t Phuang trinh (a): 7x - y - 6z + 3 = 0. n J_.10.28^ Ma M 6 ( a ) nen -2 + 3 t - 2 + 4 t - 2 + 6 t - 3 = 0=>t = —=>M U3'13'13 T H 2: (a) n OA = { l ) , suy ra I la trung diem doan OA => I 1 1 K2'2'2) 3) Tonghgfp (3 3 5^ Suy ra I B = rs 3 ^ Vi d\ 3.2,8. Viet phuang trinh mat ph5ng (a) biet: l2'~2'2j = > B C A I B = [2 2 ) la VTPTcua (a). Ph Phuang trinh (a): x + y -1 = o . 1) (a) di qua hai diem A(1;1;1),B(2;-1;3) va song song voi OC v6i C(-l;2;-l). "vTd Vi dv 3.2.9. Lap phuang trinh mat phing (P), biet: 2) (a) di qua M ( l ; l ; l ) , vuong goc voi (P): 2x - y + z - 1 = 0 va song song vol 1) (P) di qua giao tuyen ciia hai mat phSng (a):x-3z-2=0; (P):y-2z+l = 0 va khoang each tu M ( 0;0;—\ ^ 7 den (P) bang—•== . • 2 1 -3 • \j 6V3 3) (a) vuong goc vai hai mat ph3ng (P):x + y + z - l = 0 , (Q):2x-y + 3z-4 = 0 2) (P) di qua hai diem A(1;2;1),B(-2;1;3) sao cho khoang each tu va khoang each tu O den (a) bang \/26 . C ( 2 ; - l ; l ) den (P) bang hai khoang each tu D(0;3;1) den (P). 4) (a) di qua BC va each deu hai diem A , 0 . Lai gidi. Lai gidi. 1) Gia su (P): ax + by + cz + d = 0. 1) Taco O C = ( - l ; 2 ; - l ) , s u y r a A B A O C = (-2;-1;0) Taco A(2;-1;0),B(5;1;1) la diem chung cua (a) va (p) Vi (a) di qua A , B va song song voi CD nen: Vi (P) di qua giao tuyen ciia hai mat phang (a) va (P) nen A , B € (P) (a)nhan n = - A B A O C = (2;1;0) lamVTPT. 2a-b + d = 0 Jb = 2a + d Suy ra Suy ra phuang trinh (a): 2x + y - 3 = 0 . 5a + 'b + c + d• = 0 ' ^ l c = - 7 a - 2 d ' 2) Taco: n,^ = (2;-l;l), u^=(2;l;-3) l|c + d [(a)l(P) J^ltkhac: d(M,(P)) = - ^ = ^ _ Do "a ="p ^ " A =(2;8;4). (a)//A Phuong trinh (a): x + 4y + 2z - 7 = 0. ^ | c + 2d| = J^JJT^ o 27(c .2d)2 . 49(a2 . b^ ^c^) 3) Taco iv[ = (l;l;l), n^ = (2;-l;3) Ian lugtla VTPTcua (P) va (Q). Vi (a) vuong goc vai hai mat phang (P) va (Q) nen (a) nhan vec to ^ 27.49a2 = 49 a +(2a + d)2+(7a + 2d)^ n = n ^ A n ^ = (4;-l;-3) lamVTPT. I' * a.-d V - Suy ra phuang trinh (a) c6 dang : 4 x - y - 3 z + d = 0. ^ 2 7 a 2 + 3 2 a d + 5d2=o d = ±26. N/26 d ~ -a b - a;c = -5a . Suy ra phuang trinh (P) la: Vay phuang trinh (a): 4x - y - 3z ± 26 = 0. ^^ + a y - 5 a z - a = 0 o x + y - 5 z - l = 0.
Cty TNHH MTV DWH Khang Vift d = - — a => b = a;c = - — a. Suy ra p h u o n g t r i n h (P) la: yf 3.2.11. T r o n g k h o n g gian O x y z cho ba d u o n g t h a n g * 5 5 5 x = -2t 5 x - 1 7 y - 3 6 z - 2 7 = 0. x - l _ y +l _ z - l x+l _ y - l _ z va d3 : y = - i - 4 t . ".H, 2) G i a s u ( P ) : a x + b y + cz + d = 0 z = - l + 2t - b + 2c a = J) Viet p h u o n g t r i n h mat phang (a) d i qua d j va song song v o i d j . ' a+2b+c+d=0 3 ViA,Be(P)=^ _2^^^,^3^^d = 0' 5b + 5c 1) Viet p h u o n g t r i n h m|t p h a n g (P) d i qua d j va cSt d p d g Ian lu(?t tai A,B d = - sao cho A B = \/l3 . M a t k h a c : d ( C , ( P ) ) = 2 d ( D , ( P ) ) » | 2 a - b + c + d| = 2|3b + c + d 2) G p i (P) la m a t p h a n g chua d j va d3. L a p p h u o n g t r i n h m a t p h i n g (Q) o 5b-c = 2 | 2 b - c c:>c = - b , c = 3b chua d , va tao v o i mat phang (P) m p t goc o thoa cos(p = — . Vl45 • V o i c = - b , ta chon b = - 1 => c = 1 =>a = l , d = 0 Lai gidi. Phuong trinh (a): x - y + z = 0 . 1) D u o n g thang d j d i qua M ( l ; - l ; l ) c6 Uj = ( l ; 2 ; - l ) la V T C P 5 20 • V o i c = 3 b , ta chon b = 1 => c = 3,a = — , d = . d j C O U2 = ( 2 ; 3 ; - l ) la VTCP 3 3 P h u o n g t r i n h ( a ) : 5x + 3y + 9z - 20 = 0 . V i ( a ) c h u a d j va song song v o i d j nen Uj A U J = ( l ; - l ; - l ) l a V T P T o i a (a). ^^^7^^^^;^;^^;^—;^^ vahaiduong Phuong t r i n h ( a ) : x - y - z - 1 = 0 . ; _ x - 1 _ y + 1 _ z + 2 _^ , x-2 _ y + 2 _ z-1 • 2 ~ 1 ~ -1 ' ^' 1 ~ 2 " -4 • 2) T a c o A e d j => A ( l + a ; - l + 2 a ; l - a ) , B e d j = ^ B ( - 2 b ; - l - 4 b ; - l + 2b) 1) Viet p h u o n g t r i n h m a t p h a n g (P) d i qua A va d j . Suy ra A B = ( - a - 2 b - l ; - 2 ( a + 2b);a + 2 b - 2 ) , dat x = a + 2b 2) C h u n g m i n h rang d j va d j cat nhau. Viet p h u o n g t r i n h mat p h l n g (Q) Tu AB = Vl3=>(x + l ) 2 + 4 x ^ + ( x - 2 ) 2 = 1 3 o x = - l , x = | . chua d j va d j • Lot gidi. * Voi x = l=> AB = (0;2;-3), ta c6 u=(2;3;-l) la VTCP cua dj va A(-l;l;0)ed2 A€(a). Ta c6: D u o n g t h i n g d i qua M ( l ; - l ; - 2 ) , VTCP u ^ = ( 2 ; l ; - l ) Suy ra n = A B , u = (7;-6;-4) laVTPTcua (P). D u o n g t h i n g d j d i qua N ( 2 ; - 2 ; l ) , V T C P u ^ = ( l ; 2 ; - 4 ) Phuong t r i n h (P): 7x - 6y - 4z +13 = 0 . 1) T a c o : A M = ( 0 ; - 3 ; - 5 ) D o (P) d i qua A va d^ nen np = A M A Uj = ( - 8 ; 1 0 ; - 6 ) • V6ix = i=>XB = (-^;-^;-^). 3 3 3 3 i. •„ Suy ra p h u o n g t r i n h ( P ) : 4x - 5y + 3z - 3 = 0 . S u y r a ri = r - 3 A B , u 1 = (-14;11;5) l a V T P T c u a (P). l + 2t = 2 + t ' 2t-t' = l Phuong t r i n h ( p ) : 14x - l l y - 5z - 25 = 0 . 2) Xet h$ p h u o n g t r i n h _ l + t = - 2 + 2 t ' ^ t - 2 t ' = -lt = t' = l _2-t = l-4t' - t + 4t' = 3 ^) D u o n g t h i n g d j d i qua M ( l ; - l ; l ) c6 VTCP Uj = ( 1 ; 2 ; - 1 ) . 'I Suy ra d j v a d2 cat n h a u tgi E ( 3 ; 0 ; - 3 ) . ^^ong t h i n g d j d i qua E(-1;1;0), c6 VTCP u ^ = ( 2 ; 3 ; - l ) . Ta CO n g = Uj A U j = ( - 2 ; 7 ; 3 ) D u o n g t h i n g d j d i qua N ( 0 ; - 1 ; - 1 ) v a c o V T C P u ^ = ( - 2 ; - 4 ; 2 ) .•. P h u o n g t r i n h (Q): 2x - 7y - 3z - 1 5 = 0 . Taco U3 = - 2 u 4 d j //d3 . D o d o n ^ = u ^ A M N = ( - 4 ; 3 ; 2 ) . '
Cam nan)i hiycii Ihi 1)11 lliiih hoc \i;ii\fen TatThu Vi mat phSng (Q) d i qua d j nen phuang trinh (Q) c6 d^ng: M?t khac VQABC = 8 » 7OA.OB.OC = 8 abc = — ax + by + cz + a - b = 0 (1) ° 48 vol a^ + b^ + > 0 va 2a + 3b - c = 0 o c = 2a + 3b. b(2b-l) 2 b - 2 = ^ « 1 9 2 b 3 - l 6 0 b 2 + 3 2 b - l = 0 3 l4a-3b-2cl 9lb Mat khac coscp = nr 729(a2+b2+c2) 729(5a2+12ab + 10b2) b4=>a = l , c = l 4 2 6 9lb >6 o (4b - l)(48b2 - 28b +1) = 0 o Nen ta c6: b= Z ± ^ = > a - ^ ^ c 24 12 12 729(5a^+12ab + 10b2) ^145 24 12 12 a= - b 10 Vay phuong trinh mat can lap la: (a^): 6x + 3y + 2z - 1 2 = 0 ; 3V5 b = 2\l5a^ + 12ab + lOb^ « 20a2 + 48ab - Sb^ = 0 a=- =b ) : 2(5 - ^37 )x + (7 + N/37 )y + 2(1 - V37 )z - 24 = 0; 2 va (a3): 2(5 + N/37)x + (7 - N/37)y + 2(1 + V37)z - 24 = 0. • Voi a = ^ b ta chpn b = 10 => a = l,c = 32 . 2) Vi mat phling (P) d i qua M nen phuang trinh ciia (P) c6 dang: Phuang trinh (Q) la: x + lOy + 32z - 9 = 0. a(x -1) + b(y - 2) + cz = 0 o ax + by + cz - a - 2b = 0 (*) • Voi a = t3 chpn b = -2 => a = 5,c = 4 . voi a^ + b^ + c^ > 0. Phuang trinh (Q) la: 5x - 2y + 4z + 7 = 0 . Mat khac, (P) 1 (P) nen ri^.n,^ = 0 « . a + b + c = 0c = - a - b . V i dy 3.2.12. Trong khong gian Oxyz cho duong thang A c6 phuang trinh Ta c6: sin cp = |-2a + 2b + 3cl 1 5a + b| sin(uA'np)i yil^£±3 vadiem M ( l ; 2 ; 0 ) . -2 2 3 V / / ^fl7.^JJ7b^ y34(a2+ab + b2) 1) Vie't phuang trinh mat phang (a) di qua M , song song voi A va (a) tao 5a + b| Ma sincp = ^l-cos^cp = ^ nen ta c6: vai ba tia Ox,Oy,Oz mgt t u di^n c6 the tich bang 8. V34(a2+ab + b2) V34 2) Viet phuong trinh m|t phing (P) di qua M , vuong goc voi (P):x+y+z-3=0 31 va tao voi A mpt goc 9 thoa cos 9 = J (5a + b)2 = 3(a2 + ab + b ^ ) « 22a2 + 7ab - 2b2 = 0 o 34 Lai giai. 1) Gia su (a) catba true Ox,Oy,Oz Ian lugt tai> Voi a = — b , t a c h Q n b = 11 => a = 2,c =-13 Afi;0;0 0;-^;0 0;0;- voi a,b,c > 0 . b c f^en phuong trinh cua (P) la: 2x + l l y - 13z - 24 = 0 . Khi do, phuang trinh ( a ) : ax + by + cz = 1 Voi a = - l b , t a c h Q n b = -2=>a = l,c = l ' Vi (a) di qua M va song song vai A nen ta c6: •^en phuang trinh ciia (p) la: x - 2y + z + 3 = 0 = 0. fa = -2b + l a + 2b = l -2a + 2b + 3c = 0 c = -2b + - 3
-a —y • CtyTNHHmv DWH KhanxViit V i dv 3.2.13. Trong khong gian Oxyz cho hai mat phSng (P):x+2y+2z-3 = 0, Led gidi. m|it phSng (Q): 2x - y + 2z - 9 = 0 va duong thang A : ^= = Taco AB = ( - 2 ; - 2 ; - 3 ) , A C = ( l ; - 2 ; - l ) => AB A AC = (-4;-5;6) 1 1 -2 Phuong trinh (ABC): 4x + 5y - 6z - 8 = 0 . 1) Gpi (a) la m|it phang phan giac ciia goc hpp boi hai m | t phang (P) va (Q) Gpi I(x;y;z) la tam duong tron ngoai tiep tam giac ABC . Ta c6: Tim giao diem ciia duong thang A va mat phang ( a ) . 'A\ IC (x-l)2+(y-2)2+(2_l)2^(^_2)2^y2^^2 2) Viet phuong trinh mat phang (p) d i qua giao tuyen ciia (P) va (Q), dorig IB = IC o ('^ + 1)' + + (2 + 2)2 = (X - 2)2 + y 2 + ^2 thai each E(8;-2;-9) mpt khoang Ion nha't. l€(ABC) z-8 = 0 L o t gidi. r 1) Gpi M la giao diem ciia mat phang (a) va duong t h i n g A . 27 X = ^x-2y-2 =- l 154 Ta CO M € A nen M(4 +1; t; -2t) 6x + 4z = - l o< 65 V i M e (a) nentaco: d(M,(P)) = d(M,(Q)) y= 77 4x + 5 y - 6 z - 8 = 0 4+t +2t-4t-3| 2(4 + t ) - t - 4 t - 9 79 z= t - 1 = 3t + l t = 0,t = - l . 154 3 3 Tu do ta CO duoc hai diem M la: Mj(4;0;0) va M 2 ( 3 ; - l ; 2 ) . Vay I ( 27 65 79 ^ 154'77' 154 2) Ta CO A ( 3 ; - l ; l ) va B(-4;l;9) la hai diem thupc giao ciia (P) va (Q). Do do (P) d i qua giao tuyen ciia hai mat phang (P) va (Q) khi va chi khi AH.BC = 0 Gpi H ( x ; y ; z ) la tr^c tam tam giac ABC. Ta c6: A,B€(P). BH.AC = 0 (*). He (ABC) (x = 3 - 7 t Ta CO AB = (-7; 2; 8) nen phuong trinh A B : y = - l + 2t, t e : Ma A H = ( x - l ; y - 2 ; z - l ) , BC = (3;0;2), BH = ( x . l ; y ; z . 2 ) , AC = ( l ; - 2 ; - l ) z = l + 8t 127 X = Gpi K la hinh chieu ciia E len AB, suy ra •3(x-l) + 2(z-l) = 0 ^3x + 2 z - 5 = 0 77 Nen (*) E K . A B = 0 -7(-5-7t) +2(1+ 2t) +8(10+ 8t) = 0 < » t = - l 2 z= 77 Suyra K ( 1 0 ; - 3 ; - 7 ) , E K = (2;-1;2). '127 2 4 ^ ^ V^yH Gpi H la hinh chieu ciia E len mat phang (p), khi do: d(E,(P)) = E H < E K 77 '77'77 , Suy ra d(E,(P)) Ion nhat khi va chi khi H = K hay (p) la m^t phSng di q"^ ^* dy 3.2.15. Trong h^ tpa dp Oxyz , cho hai duong thang I K va vuong goc voi E K . d, : i l ^ = I ^ = i l ^ . d , :f = ^ =^ vadiem A ( - 2 ; l ; 0 ) . Phuong trinh (p): 2x - y + 2z - 9 = O . (ta thay (p) = (Q)). Chung minh A , d j , d 2 C u n g nam trong mpt mat phang. T i m tpa dp cac V i d\ 3.2,14. Trong khong gian Oxyz cho ba diem A ( l ; 2; I ) , B ( - l ; O; -2)' pinh B,C cua tam giac ABC biet duong cao tir B nam tren dj va duong C(2; 0; 0). Tim tpa dp tam duong tron ngoai tiep va true tam tam giac A B C J ^ phan giac trong goc C n i m tren d j .
Cam nana l,n,cn Ihi DIl lln.h ho. N^micnTatlhu Cty TNHH MTV DWH Khang Viet Lai gidi. Vi dv 3.2.16. Trong khong gian Oxyz cho duong thing Duong thang d, di qua M(l;2;4) vac c6 VTCP = (1;1;1) • x-4m4-3 _ y - 2 m - 3 _ z - 8m - 7 Duong thang di qua N(0;3;2) vac c6 VTCP U2 ={V,-V,2). 2m-l m +1 4m + 3 voi m «£^ ' 4 ' 2 Chung minh rSng khi m thay doi thi duong thSng d^ luon nkm trong Gpi I la giao diem cua d p d 2 1(1; 2; 4). Mat phing (a) chua dj,d2 c6 n„ = U i , U 2 = (3;-l;-2) va di qua I nen mpt mat phang co dinh. Vie't phuong trinh mat phang do. I Lai gidi. phuong trinh : 3 x - y - 2 z + 7 = 0. Duong thang d^ diqua A(4m-3;2m + 3;8m + 7) va c6 VTCP Ta tha'y A e (a). V|y A,dj,d2 cung thuQC mp (a). u = (2m - l;m + l;4m + 3) Xac dinh diem C: Gia su d^ c (a): ax + by + cz + d = 0 voi mpi m , khi do ta c6: Gpi (P) la mp di qua A va vuong goc voi dj=:> (P):x + y + z + l = 0. a(2m -1) + b(m +1) + c(4m + 3) = 0 Co C = (P) n (d2) nen tpa dp diem C la nghi^m ciia h^ phuong trinh " a(4m - 3) + b(2m + 3) + c(8m + 7) + d = 0 Vm x+y+z+l=0 f (2a + b + 4c)m - a + b + 3c = 0 , x _ y - 3 _ z - 2 => C(-3;6;-4) |(4a + 2b + 8c)m - 3a + 3b + 7c + d = 0 Vm , 1 ~ -1 ~ 2 '2a + b + 4c = 0 Gpi A' la diem do'i xung cua A qua d2 . -a + b + 3c = 0 b-lOa Ke A H l d 2 ^ H ( t ; 3 - t ; 2 + 2t) AH = (t + 2;2-t;2 + 2t) 2a + b + 4c = 0 c = -3a , AH.u^ = 0«(t + 2 ) - ( 2 - t ) + 2(2 + 2t) = 0=>t = - | d = -6a -3a + 3b + 7c + d = 0 ' 2,11.2^ ^2 19 4^ Ta chpn a = 1 => b = 10,c = -3,d = -6 • H ' 3 ' 3 '3 • A' 3' 3 ' 3 Vay d^ c ( a ) : x + 1 0 y - 3 z - 6 = 0. Co duong thang BC la duong thSng BA' di qua C va c6 VTCP Vi du 3.2.17. Trong khong gian Oxyz cho bon diem A(l;l;l),B(-l;0;-2), C(2;-l;0),D(-2;2;3). u = O V = -(ll;l;16) chpn u' = (ll;l;16) 3 1) Chung minh rang A,B,C,D la bon dinh ciia mpt tu di^n va tinh the tich hi dif n ABCD . Suy ra phuong trinh BC: 1 1 16 1 Tpa dp diem B la nghi?m cua h? phuong trinh : 2) Lap phuong trinh mat phang (a) song song voi AB,CD va c5t hai duong I _ yj-6 ^ z+_4 thang AC,BD Ian lupt tai hai diem M,N thoa = AM^-l.i 11 1 ' 16 ^ B lr29 34 44^ x ^ ^ y-2^^ Zj-4 5'5' 5 l,AM^ 1 " 1 1 Gpi G la trpng tam ciia tu dif n ABCD, (P) la mat phSng di qua G cat cac Ccinh AB,AC, AD Ian lupt tai B',C',D'. Viet phuong trinh mat phang (P) Vay B '29 34 44' ;C(-3;6;-4). ^„^'t tu d i g n AB'C'D' CO the'tichLaiIongidi.nhat. n. 1 U ' s ' s J ^ c6: A B = (-2; -1; -3), A C = (1; -2; -1), A D = (-3; 1; 2) ra A B A C = (-5;-5; 5) ^ ( A B A C ) . A D = 20 ^ 0 A A
M.LiAm miv wvH Kltang Vift ^latkhac^^^^^=A^li^C^AD:^27 27^, 45 VABCD AB.ACAD ~ 64 ^ A B C D ' ^ ^ V^BCD = — • Do do A, B,C, D la bon dinh cua tu di#n. m 10 Thetichtudi?n A B C D la: Vf^^^-^ = - {AS A AC).AD\ DSng thuc xay ra khi va chi khi ~ = - i l £ = ^1 i 6 AB' A C AD' 4' DO do: V^g.^.p. nho nhat « ( a ) song song voi mat ph3ng (BCD). 2) Taco B D = (-1;2;5) nen B D = N/30,AC = >/6, Ta c6: BC = (3; - 1 ; 2), BD = (-1; 2; 5) n„ = BC A BD = (-9; -17; 5) C D = (-4; 3; 3) => A B A C D = (6; 18;-10). Vay phuong trinh mat phang (a) la: 9x + 17y - 5z - 6 = 0. Vi (a) song song vol A B , C D nen n^ =(3;9;-5). Vi M N , A B , C D Ian lugrt nam tren ba mat phing song song gom : M^t * Bai tap van dyng phang di qua A B va song song voi C D , mat ph3ng di qua C D va song song Bai 3-2'l' Viet phuang trinh mat phang (a), biet: voi A B va mat phang (a) nen theo dinh ly Talet dao trong khong gian ta x = 2t BN AM BN AM /-^^^ „^,2 cAx-r2 suy ra: = - f = - = B N = V5AM o BN'^ = SAM''. 1) (a) di qua diem A(2;3;-l) va duong thing dj : • y = 1 - 1 . BD AC VSO N/6 [z = 2 + t AM^ - 1 o AM^ - AM^ = BN^ = 5AM^ ^ AM^ = 6. Mat khac: 2) (a) chua hai duong thing dj va dj : ^^-^ = ^= . ^AMj x=l+t 3) (a) chua dj va song song voi Oy. Ta CO phuong trinh AC: y = l - 2 t = > M { l + t ; l - 2 t ; l - t ) :0. Huong dan giai z= l-t 1) Duong thing dj di qua B(0;l;2), VTCP Ui' = (2;-l;l) Suy ra AM = (t; -2t; -t) ^ AM^ - 6 o t = ±1. Suy ra AB = (-2;-2;3), n„ =ABAUJ =(1;8;6). . V a i t = l,tac6M(2;-l;0)HCnentalo9itru6ngh(?pnay. Phuong trinh (a): x + By + 6z - 20 = 0. . Voi t = - 1 , ta CO M(0; 3; 2) nen phuong trinh (a): 3x + 9y - 5z -17 = 0. 2) Duong thing 62 di qua C(-l;2;0), VTCP u^ = (-2;l;-l) ^ 1 1 Suy ra BC = (-l;l;-2) => n„ = U j A BC = (1;3;1) 3) Ta CO G 2 2 Phuong trinh (a): x + 3y + z - 5 = 0. i l l GQi Aj la trQng tam cua tam giac ABC, suy ra A, ^) Oy CO VTCP ic = (0;1;0). Suy ra = u^ A k = (-1;0;2) '3'3'3j L AAi = 1 _2 _2 . Tu do ta c6: Phuong trinh (a): x - 2z =: 0 . Suy ra AG = '3' 3' 3 3.2.2. Trong khong gian Oxyz cho 4 diem: 2' 2j AB AC AD A(l;l;l), B(-l;2;0), C(-2;0;-l),D(3;-l;2). SG = - A A ; = i ( A B + AC + AD) = i ^AB'' AB' + AC- A C + AD' AD' Viet phuong trinh mat phing (ABC). Tim tpa do trvrc tam tam giac ABC. 4-" ' 4^ ' 4 ^) Vi^'t phuong trinh mat phing (a) di qua AB va song song voi CD. Vi B',C',D',G dong phang nen ta c6: Viet phuong trinh mat phing (P) di qua AB va each deu C, D. ' ' AB AC AD A = !=> AB + AC + AD = 4. + +• AB' A C AD' ^' Vig't phuong trinh mat phing (P) di qua BC va each A mpt khoang Ion nhat. 4 ^AB' A C A D ' j ^ Huong dan giai ' "^aco AB = (-2;l;-l),AC = (-3;l;2)=^riABC =(3;7;1) AB AC AD
Cam nangTIyfit thi E)IJ H ' " " '«?C(2c;-c;l-2c), D 6 d j = > D ( l - 2 d ; - l + d;-2 + 2d) Do B g (a) nen suy ra -2a + b - c = 0 => c = -2a + b CD = (-2x + l ; x - l ; 2 x - 3 ) , x = d + c Nen ta viet lai phuong trinh (p) nhu sau: ax + by + (-2a + b)z + a - 2b = 0 Suyra CD = 738 o ( 2 x - l ) 2 + ( x - l ) 2 + ( 2 x - 3 ) ^ =38 Vi (a) each deu C D nen d(C,(a)) = d(D,(a)) « 9 x ^ - 1 8 x - 2 7 = 0 o x = -l,x = 3 la - 3b| = |b| o a = 4b, a = 2b . Voi x = -1 =:> CD = (3; -2; -5), AB = (-2; -2; -1) • Voia =4b, ta chQn b = 1 => a = 4 . Phuong trinh (P):4x + y - 7 z + 2 = 0 Suy ra n^ = AB A CD = (8;-13;10) • Vol a = 2b, ta chpn b = 1 => a = 2 . Phuong trinh (p): 2x + y - 3z = 0. t Phuong trinh (a): 8x - 13y + lOz -12 = 0 x=-l +t f/oi x = 3=oCD = (-5;2;3).Suy ra n^ = AB A CD = (-4;ll;-14) 4) Ta C O phuong trinh BC: • y = 2 + 2t z=t Phuong trinh (a): 4x - l l y + 14z - 24 = 0. 2) Vi (P) song song voi hai duong thing dpdg nen ta c6 GQ'I K la hinh chieu cua A len BC, suy ra K ( - l +1; 2 + 2t; t) np = u i A U 3 =(l;-4;3) AK = (t-2;2t + l ; t - l ) . 11 4 Phuong trinh (P) c6 dang: x - 4 y + 3z + m = 0 1 Do AK.BC = 0=>t = - = > AK = Lay E(0;0;l)edi, F(-1;0;-1) 6 dj 6 ' 6'3' "6. Gpi H la hinh chieu cua A len mat phang (P), ta c6 A H = d(A,(P)), Vi (p) each deu hai duong thing d^dg nen ta c6: va AH < AK nen d(A,(P)) Ion nhat khi va chi khi H = K • j E , ( P ) ) = d(F,(P))o m + 3|= m - 4
Cam nang luycit th! Till Iliiih hoc - iwguyen i aTTmr- Do MN//(Q) nen MN.nQ = 0 Vay CO hai m$t phSng (a) can tim la 2(n + 2m - 2) + (n - m +1) + (n - 2m +1) = 0 m = 2 - 4n i r (a): X + 3z +1 = 0 hoac (a): 3x + 4y - z + 5 = 0. Matkhac: MN = Vl4 o (n + 2m - 2)^+(n - m +1)^ + ( n - 2 m + 1)^ =14 3) Mat phing (a) di qua diem C(-l;0;2) nen c6 phuong trinh dang C^155n2-92n = 0 » n - 0 , n = — . A(x + l) + By + C(z-2) = 0, A^+B^+C^ >0. 155 .'Jii-; Vi (a) qua D(l;-2;3) nen 2A -2B + C = 0 C = 2B-2A (1). Do X K , < - - nen ta CO n = 0 =^ m = 2 M(-3;1;2),N(-1;0;-1) Taco d(0,(a)) = 2 nen A-2C| - = 2 (2). Suy ra Hp = OM A ON = (-1; -5; 1) VAVBVC^ Phuong trinh (P): x + 5y - z = 0 . The (1) vao (2) roi binh phuong, riit gpn ta thu dug-c Bai 3.2,4. Lap phuong trinh mat phSng (a) biet rA = 2B 5 A 2 - 8 A B - 4 B 2 =0c^ 2 . Do A ^ + B^ + > 0 nen 1) (a) qua hai diem A(l;2;-l),B(0;-3;2) va vuong goc vai m^it phang A = --B 5 ( P ) : 2 x - y - z + l = 0. * Neu A = 2B thi chpn B = 1 => A = 2,C = -2, do do phuong trinh mat phMng 2) (a) each deu hai mat phSng ((3): x + 2y - 2z + 2 = 0, (y): 2x + 2y + z + 3 = 0. i|:(a) la 2x + y - 2 z + 6 = 0. 3) (a) qua hai diem C(-l;0;2),D(l;-2;3) va khoang each tii goc tQa dp toi mat phang (a) la 2. »Je'u A = -|B thi ch(?n B =-5 => A = 2,C =-14, do do phuong trinh mat 4) (a) song song voi mat phang (Q): x - 2y - 2z - 3 = 0 va khoang each giiia phMng (a) la 2x - 5y - 14z + 30 = 0. hai mat phang la 3. Vay CO hai mat ph5ng thoa man 2x + y - 2z + 6 = 0, 2x - 5y - 14z + 30 = 0. \ 5) (a) di qua E(0; 1; 1) va d(A,(a)) = 2;d(B,(a)) = y , trong do A(l;2;-1), 4) (a) song song voi mat ph^ng (Q): x - 2y - 2z - 3 = 0 nen c6 phuong trinh B(0;-3;2). (a): X - 2y - 2z + D = 0. Lay diem N(3; 0; 0) 6 (Q). Huong dan giai 3+D Taco d((a),(Q)) = d ( N , ( a ) ) o = 3 1) Taco A B ( - l ; - 5 ; 3 ) , n ( p ) ( 2 ; - l ; - l ) nen AB, ri( P ) = (8;5;11). Mat phing (a) qua A, B va vuong goc voi mat phang (P) nen « | D + 3 = 9 C ^ D = 6;D = -12. "(o) ^ AB, 1 n(p) ^ n(„) =[AB, ii(p)J = (8;5;ll). Vay CO hai mat phang can tim x - 2y - 2z + 6 = 0, x - 2y - 2z -12 = 0. Phuong trinh mat phang (a) can tim Mat phang (a) qua E(0; 1; 1) c6 phuong trinh dang 8 ( x - l ) + 5(y-2) + ll(z + l) = 08x + 5y + l l z - 7 = 0. Ax + B{y -1) + C(z -1) = 0, A^ + B^ + > 0. 2) Ggi M(x;y;z) la mpt diem bat ky thuQC mat phMng (a). Theobai ra d(A,(a)) = 2;d(B,(a)) = y voi A(l;2;-l),B(0;-3;2) nen x + 2y-2z + 2 2x + 2y + z + 3 Taco d(M,(p)) = d(M,(y))c^ A + B - 2C| ^1^+2^+(-2)2 72^+2^+12 =2 X + 2y - 2z + 2| = |2x + 2y + z + 3 A^ + B^+e A + B-2C| = 2/A2 + B2 + C2 (!) -4B + Ci 11 x+2y-2z+2=2x+2y+z+3 x+3z+l=0 n | A + B-2C| = 14|-4B + C (2) I x + 2y - 2z + 2 = -2x - 2y - z - 3 3x + 4y - z + 5 = 0
l i V M H M i V DVVH Kliang Vi(t Huong dan giai -67B + 36C A=i Ivlat cau (S) CO tam 1(2;-3; 1), ban kinh R = N/42 11(A + B - 2 C ) = 14(-4B + C) 11 Tu (2) ta CO 11(A + B - 2 C ) = 14(4B-C) 45B + 8C puong thiing d, di qua Mi(-5;1;-13), VTCP u^ = (2;-3; 2) 11 puong thang dj di qua M 2 ( - 7 ; - l ; 8 ) , VTCP u^ = (3;-2;l) -67B + 36C thay vao (1) ta c6 phuong trinh: ,) Vi mat phSng (P) song song voi hai duong thSng d p d j nen ta c6: • Voi A = 11 Hp =ui A U j =(1;4;5) -67B + 36C '-56B + 1 4 c f = 4 + B2+C2 Suy ra phuong trinh (P) c6 dang: x + 4y + 5z + m = 0. 11 11 M|t khac, (P) tiep xiic voi (S) nen d(I, (P)) = R . » 38265^ - 4432BC + 1368C2 = 0 (3) |m - 5 Phuong trinh (3) chi c6 nghif m B = C = 0, khi do A = 0 (khong thoa man = >/42 m - 5 = 4 2 o m = 47,m = -37. 742 dieuki?n A 2 + B 2 + C 2 > 0 ) ^Pr^y phuong trinh (P) la: x + 4y + 5z - 37 = 0 ho|ic x + 4y + 5z + 47 = 0 . Voi A = thay vao (1) ta c6 phuong trinh 2) Vi mat phang (a) chua dj nen phuong trinh (a) c6 dang 11 a(x + 5) + b(y -1) + c(z +13) = 0 o ax + by + cz + 5a - b + 13c = 0. N2 56B-14C 45B + 8 C ' + B 2 + C 2 =4 11 11 . Voi a^ + b 2 +c2 ^0 va 2a - 3b+ 2c = 0 o b = ^ ^ ^ ^ . 3 1362B^ +1112BC + 136C^ =0oB = --C, B = - — C . Taco: sincp = yJ\-cos(i> ^"Y " '^°s("a'"2) 3 227 2 +) Voi B = — C thi chpn C = -3=>B = 2,A = 6 phuong trinh mat phang (a) |3a-2b + c 3 - . "2 ^ Vli.Va^ + b^+c^ 77,. la 6x + 2 y - 3 z + l = 0. | +) Voi B = = - ^ C thi chpn C = 227 =:> B =-34, A = 26 phuong trinh (a) la o T 2a + 2c 2a + 2 c f 2 3a - 2 . +c a2 + ' +c 26x - 34y + 227z -193 = 0. Vay c6 hai mSt phSng can tim la I 6x + 2 y - 3 z + l = 0, 26x-34y + 227z-193 = 0. (5a - cf = 2(13a^ + Sac + ^c^) o a^ + 26ab + 25c^ = 0 Bai 3.2.5. Trong khong gian Oxyz cho m|t cau (S):x^+y^+z^-4x+6y-2z-28=fl *^a = -c,a = -25c. va hai duong thang d^: • 2 -3 2 ' ^ 3 -2 1 Voi a = - c , chQn c = - l , a l,b = 0 . Phuong trinh (a): x - z - 8 = 0 . * Voi a = -25c, ch(?n c = - l , a = 25,b = 16. 1) Viet phuong trinh mat phJing (P) tiep xiic voi mat cau (S) va song song ^ W n g trinh (a): 25x + 16y - z + 96 = 0. hai duong thang d j ; d2 . 2) Viet phuong trinh mat phing (a) di qua dj va t^o voi d j mpt goc (p ^' (P) chua d2 nen phuong trinh c6 dang: ax + by + cz + 7a + b - 8c = 0 N/42 cos(p = - ^ . ^oi +b^ + ^0 va 3 a - 2 b + c = 0=>c = 2 b - 3 a . 3) Viet phuong trinh mat phing ((3) chua d j va cat (S) theo mpt duong tron' T h p , . . 124 N/6 '^o gia thiet bai toan, ta c6: d(I,(P)) = .42 -
Cty TNHH MTV DWH Khanx Viet l9a - 2b - 7cL ^ ^ ^ 3 30a - 16b = ^edOa^ - 12ab + Sb^) OA = OB • a 2 + b 2 + c 2 = 3 2 7^ b^+c OA = AB OA = AB (a-4)2+(b-4)2+c2=32 3(15a-8b)^ =2(10a^ -12ab + 5b^) b 2 - 4 b = 0=e>b = 0,b = 4 . D o d 6 B(4;0;4) hoac B ( 0 ; 4 ; 4 ) . Phuong trinh mat phang (P) la: x + 2y + z +1 = 0 . • B(0;4;4), taco OA, OB = (16;-16; 16) nen phuong trinh (OAB): • Voi a = — b , c h Q n b = 670,a = 379,c = 203. x-y+z=0. 670 • B(4;0;4),tac6 OA, OB = (l6;-16;-16) nen phuong trinh (OAB): Phuang trinh (P) la: 379x + 670y + 203z +1699 = 0. x-y-z=0. Bai 3.2.6. Trong khong gian vol hf true tpa dp Oxyz, cho 2 diem A(2;0;]), Bai 3.2.8. Trong khong gian h^ toa dp Oxyz, cho duong thSng B(0;-2;3) va mat ph^ng ( P ) : 2 x - y - z + 4 = 0. Tim tpa dp diem M thuoc X — 2 y +1 z (P) sao cho M A = M B = 3 . A: = —— = — va mat phang (P):x + y + z - 3 = 0. Gpi I la giao diem Huong dan giai ciia A va (P). Tim tpa dp diem M thupe (P) sao cho MI vuong goc voi A Gpi E la trung diem A B ta c6: E(l;-1;2), A B = (-2;-2;2) va MI = 4V14 . Phuang trinh mat phang trung true (Q) eiia A B c6 phuong trinh: Huong dan giai c=3+-a x+y-z+2=0. 2a-b-e+4=0 2 Taco A eat (P) tai I(l;l;l). Goi M(a;b;e) suy ra: I a + b - c + 2 = 0 ': Vi M A = M B nen suy ra M € ( Q ) M € (P) bn =( Ql +) i a Diem M ( x ; y ; 3 - x - y ) € ( P ) => MI = (l - x ; l - y ; x + y - 2 ) 2 Duong thing A eo a = (l;-2;-l) laVTCP Mat khae: MA^ = 9 (a - 2)^ + -^a 2 + 1 + (|a + 2)2=9.j Ta CO : MI.a = 0 y=2x-l MI^ =16.14 i ( l - x ) 2 + ( l - y ) 2 + ( _ 2 + x + y ) 2 = 16.14 Giai ra ta dupe a = 0,a = -— x = -3 . fx = 5 hoac y = 9 f_6 4 12] y = -7 V$y CO hai diem thoa yeu cau bai toan la: M(0;1;3), M 7'7'7 . CO hai diem thoa yeu cau bai toan: M(-3;-7;13) va M(5;9;-ll). Bai 3.2.7. Trong khong gian voi h^ true tpa dp Oxyz, cho mat cau (5) ' 3.2.9. Trong khong gian tpa dp Oxyz, cho duong thang A : " ~ ~ - ~" = ~Y' phuong trinh x^+y^ + z ^ - 4 x - 4 y - 4 z = 0 v a diem A(4; 4; 0). Viet phu^"^^ mat phang ( P ) : x - 2 y + z = 0. Gpi C la giao diem ciia A voi (P), M la trinh mat phSng (OAB), biet B thupc (S) va tam giac OAB deu. Huong dan giai ^'^'m thupc A . Tinh khoang each tu M den (P), biet MC = \/6 . , ) , iv Xet B(a;b;c). Vi tam giae AOB deu nen ta c6 h^: ' '"'^ '
Cam tiattg luyftt thi tJH Htnh hqc - Nguyen lai imi ^ H u o n g dan giai H u o n g dan g i a i x = l+2t M a t p h i n g (P) c6 n ^ = C^'^'^) 1^ ^ T P T , m p ( Q ) c6 n ^ = (1;-1;1) la VTPT. Cach 1: P h u o n g t r i n h t h a m so' ciia A : < y = t ,t e R . (R)l(P) , — Ir • z = -2-t Doi => m p ( R ) CO n^ =— np,np, = (1;0;-1) la V T P T (R)l(Q) ^ ' ^ 2l ^ Q Thay x , y , z vao p h u o n g t r i n h (P) ta d u p e : Suy ra ( R ) : x - z + m = 0 . l + 2 t - 2 t - t - 2 = 0ci>t = - l = > C ( - l ; - 1 ; - l ) . m Ta CO d ( 0 ; ( R ) ) = 2 » = 2 m = tly/l. D i e m M e A » M ( l + 2 t ; t ; - 2 - 1 ) => M C = N/6 o ( 2 t + 2)^ + ( t + l ) ^ + ( t + l ) ^ =6 Vl + 0 + 1 Vay (R):x-z±2V2=0. t = 0 ^ M ( l ; 0 ; - 2 ) => d ( M ; ( P ) ) = ^ p^i 3.2.12. T r o n g k h o n g gian v o i h f to? dp O x y z , cho t u d i ^ n A B C D c6 cac t = -2 r> M ( - 3 ; - 2 ; 0) =^ d ( M ; (P)) = d i n h A ( 1 ; 2 ; 1 ) , B ( - 2 ; 1 ; 3 ) , C ( 2 ; - 1 ; 1 ) va D ( 0 ; 3 ; 1 ) . Vie't p h u o n g t r i n h m ^ t p h i n g (P) d i qua A , B sao cho khoang each t u C d e n (P) b a n g k h o a n g Cach 2: D u o n g t h i n g A c6 u = ( 2 ; l ; - l ) la V T C P each t u D den ( P ) . M a t p h i n g (P) c6 ri = (1;-2;1) la VTPT H u o n g dan g i a i Mat p h i n g (P) thoa m a n yeu cau bai toan t r o n g hai t r u o n g h p p sau: G p i H la h i n h chie'u cua M len ( P ) , suy ra c o s H M C = cos u , n j nen ta c6 T r u o n g h p p 1: (P) d i qua A , B song song v a i C D . 1 d(M,(P)) = M H = M C c o s H M C = Ta CO A B = ( - 3 ; - l ; 2 ) , C D = ( - 2 ; 4 ; 0 ) , suy ra n = r A B , C D l = ( - 8 ; - 4 ; - 1 4 ) la Bai 3.2.10. T r o n g k h o n g gian tpa d o O x y z , cho cac d i e m A(1;0;0), B(O;b;0) VTPT cua (P). P h u a n g t r i n h (P): 4x + 2y + 7z - 1 5 = 0 . C(0; 0; c ) , t r o n g d o b, c d u a n g va mat p h a n g ( P ) : y - z + 1 = 0 . Xac djnh b T r u o n g h g p 2: (P) d i qua A , B va cat C D tai I , suy ra I la t r u n g d i e m cua va c, bie't m a t p h i n g ( A B C ) v u o n g goc v o i mat p h i n g (P) va k h o a n g each CD D o d o I ( 1 ; 1 ; 1 ) = > A I = ( 0 ; - 1 ; 0 ) . ^ t u d i e m O den mat phSng ( A B C ) bang —. Vec t o p h a p t u y e n cua mat phang (P): n = A B , A I ==(2;0;3). 3 P h u a n g t r i n h ( P ) : 2x + 3z - 5 = 0 . H u o n g dan g i a i Vay ( P ) : 4 x + 2y + 7 z - 1 5 = 0 hoac ( P ) : 2 x + 3 z - 5 = 0 . Phuang trinh (ABC): j + ^ + - = 1 Sai 3.2.13. T r o n g k h o n g gian v o i h f tpa dp O x y z . Viet p h u a n g t r i n h m a t p h i n g (P) qua A(0;0;3), M ( l ; 2 ; 0 ) va c i t cac true Ox, O y Ian l u p t tai B, C sao cho Vi (ABC)l(P)^i-- =0 « b = c ^ ( A B C ) : b x + y + z - b = 0. b c tarn giac A B C c6 t r p n g t a m thupc d u o n g t h i n g A M . 1 .: 1 H u o n g dan g i a i M a d ( 0 , ( A B C ) ) = i=> Ji= = ^b = i (do b > 0 ) . y z-3 Ta CO A M ( 1 ; 2 ; - 3 ) nen p h u o n g t r i n h cua A M la: — = — = — 1 2 ~3 Vay b = c = ^ la gia trj can t i m . Gpi tpa d p d i e m B ( a ; 0 ; 0 ) , C ( b ; 0 ; 0 ) , a b ;t 0 . Bai 3.2.11. T r o n g k h o n g gian toa dp O x y z , cho hai mat phang (P): x + y + z - 3 Trpng tam t a m giac A B C la: G i>l va ( Q ) : x - y + z - l = 0 . Viet p h u a n g t r i n h m a t phSng (R) v u o n g goc 13'3' (P) va (Q) sao cho k h o a n g each tir O den (R) bMng 2.
C f y TNHH MTV DWH KhanVvu^ Vi G e A M nen - = - = - a = 2;b = 4 . ^, | C H l A B ^ | A B . C H = 0 ^ | 2 a - 3 b - c + 5 = 0 3 6 3 [BHIAC BH.AC = 0 |2a + b + c - 3 = 0 (2) X y z Khi do, phuang trinh (P) CO dang: — + — + — = 1.. ; Tu (1) va (2) suy ra a = 0;b = l;c = 2 . 2 4 3 VayH(0;l;2). Bai 3.2.14. Trong khong gian voi tpa dp Oxyz cho A(2;5;3) va duoria o 2) Giasu M ( a ; b ; c ) € ( P ) ^ 2 a + 2b + c - 3 = 0 (3) X — l y z —2 thang d : ^ = Y = —^— • Tim tpa dp hinh chieu vuong goc cua A len d va MA^" = MB^ Doi ^ - 2 b - 4 c + 5 = -4a + 4 b - 2 c + 9 2a-3b-c = 2 viet phuang trinh mat phang (P) chua duong thang d sao cho khoang MB^ = MC^ -4a + 4b - 2c + 9 = 4a - 2c + 5 (4). 2a-b = l each t u A den (P) Ion nha't. Tu (3) va (4) ta tim dupe: a = 2;b = 3;e -7 . Huong dan giki Vay M(2; 3;-7) la diem can tim. ,,3, • Duong thSng d c6 u ^ = (2;1;2) la VTCP. Bai 3.2.16. Tim m , n de 3 mat phang sau ciing d i qua mpt duong thang: Gpi H la hinh chieu cua A len d ( p ) : x + my + n z - 2 = 0, ( Q ) : x + y - 3 z + l = 0 va ( R ) : 2x + 3y + z - l = 0. =>H(1 + 2 t ; t ; 2 + 2t)r^ A H = (2t - 1; t - 5 ; 2 t - 1) Khi do hay viet phuang trinh mat ph^ng (a) d i qua duang thiing chung do Do A H I d =^ A H . i ^ = 0 o 2 ( 2 t - l ) + t - 5 + 2 ( 2 t - l ) = 0 H ( 3 ; l ; 4 ) . va tao voi (P) mot goc cp sao cho cos(p = —^2=. • Gpi H ' la hinh chieu cua A len mp(P). V679 Khi do, ta c6: A H ' < A H d(A, (P)) Ion nha't Huong dan giai o H = H ' c:> (P) 1 A H Xet h? phuong trinh: 'x + y - 3 z + l = 0 2x + 3y + z - l = 0 Suy ra A H = (1; - 4 ; 1) la VTPT cua (P) va (P) di qua H . * Cho z = l = ^ x = 6,y = -4=> A(6;-4;l)£(Q)n(R). Vay phuong trinh (P): x - 4 y + z - 3 = 0. Bai 3.2.15. Trong khong gian voi h? tpa dp Oxyz, cho ba diem A(0;l;2) * Cho z = 0=>x = - 4 , y = 3 = > B ( - 4 ; 3 ; 0 ) e ( Q ) n ( R ) . B(2;-2;l), C(-2;0;l). Ba mat phang da cho cung di qua mpt duang thang A , B G (P) 1) Viet phuong trinh mat phSng di qua ba diem A , B , C va t i m tpa dp true tam I-4m + n = -4 fm = 2 3m = 6 jn=4 tam giac A B C . 2) Tim tpa dp ciia diem M thupc mat phSng (P): 2x + 2y + z - 3 = 0 sao cho Ta c6: n = ( l ; 2; 4) la VTPT ciia (P) M A = M B = M C . Vi (a) di qua A nen phuong trinh ciia (a) c6 dang: Huong dan giai a ( x - 6 ) + b(y + 4) + c ( z - l ) = 0 . ,f 1) Taco: A B = ( 2 ; - 3 ; - l ) , A C = ( - 2 ; - l ; - l ) ^ [ A B , A C ] = (2;4;-8) la mot VTPT J^o B e (a) nen ta c6: c = -10a + 7b . ciia mp(ABC). Phuong trinh mp( A B C ) :x + 2 y - 4 z + 6 = 0. Suy ra V = (a; b; -10a + 7b) la VTPT ciia (a) Gpi H ( a ; b ; c ) la true tam tam giac A B C KJ, , , n-v -39a + 30b| = > H e ( A B C ) = > a + 2 b - 4 c + 6 = 0 (1) ^' '^cn theo gia thiet ta co: cos(p = — — n V Taco: C H = ( a ; b - l ; c - 2 ) , BH = ( a - 2 ; b + 2 ; c - l ) ^yracos(p= -39a + 30b 23