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Cam nang luy?n tht VH tltnh hqc - Nguyen Tat Thu «-ty ' ( 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 AI^ = BI^ AT = CI <^ AI^ = CI^ I e (ABC) Giai AB,AC .AI = 0 ta dupe I 14 61 1^ 30 3^ J5 6x + 2y + 2z = 9 lz = l - x x + 2y + 3z = 4 y = 16-5x x - 8 y + 5z = -17 z= l-x y = 16-5x (x-5)2+(13-5x)2+(2-x)2=l8 3x2-16x +20 = 0 Giai phuong trinh 3x^ - 16x + 20 = 0 ta dupe. ' 2 - 1 + 1 3 + 2 + 1 1 + 0-2" nen HG = 2GI, tuc la ba diem G Ta c6: AS(x - 5; y - 3; z +1), BS(x - 2; y - 3;z + 4),CS(x - 1 ; y - 2;z) SA,SB,SC doi mpt vuong goc khi va chi khi Bai 3.1.6. Cho tam giac deu ABC c6 A ( 5 ; 3 ; - l ) , B { 2 ; 3 ; - 4 ) va diem C nam trong AS.BS = 0 ( x - 5 ) ( x - 2 ) + ( y - 3 ) 2 + ( z + l)(z + 4) = 0 • BS.CS = 0 « < ! ( x - 2 ) ( x - l ) + ( y - 3 ) ( y - 2 ) + (z + 4)z = 0 CS.AS = 0 ( x - l ) ( x - 5 ) + ( y - 2 ) ( y - 3 ) + z(2 + l ) = 0 mat phSng (Oxy) c6 tung dp nho hon 3 1) Tim tpa dp diem D biet ABCD la t u di^n deu. 2) Tim tpa dp diem S biet SA,SB,SC doi mpt vuong goc. x^+y^+z^ - 7 x - 6 y + 5z = -23 Huong dan giai Vi C 6 ( O x y ) nen C(x;y;0). x ^ + y 2 + z 2 - 3 x - 5 y + 4z = -8 Ta CO AB(-3; 0; - 3), AC(X - 5; y - 3; 1), BC(x - 2; y - 3; 4) x2+y2+z2-6x-5y + z = - l l Tam giac ABC la tam giac deu nen AB = AC = BC, do do y = 5z + l l (X - x=l x = l;y:=2" V i C CO tung dp nho hon 3 nen C(l;2;0). 1) Gpi D(x;y;z). x + y - 4 z = 12 - 3 x - 3 z = -3 3z2+10z + 8 = 0 Gia 4 x = l;y = o x= l-z =18 5)2 + ( y - 3f +1^ = (X - 2f + ( y - 3)^ + 4^ (y-3)2=l "3;-3] 2) Gpi S(x;y;z). Do do H G ' A ; ^ , 0 , G I ^ ; - i - ; 0 15 30 15 15 H , I nam tren mpt duong thang. AC = BC 10- Vay tpa dp cac diem D la D(2; 6; -1) hoac of — ; 3 (x-5)^+(y-3)^+l^ x-2 X = 4) Trpng tam G cua tam giac ABC c6 tao dp thoa man f AC = AB UVVH Kharig Vi(t 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 A I = BI miv Tam giac ABC la tam giac deu nen ABCD la t u di?n deu khi va chi khi ^ . 2 9 _ 1_ 15'15' 3 Giaihe taduac H it\Tin phuong trinh 3z2 + lOz + 8 = 0 ta dupe z = - 2 z = - i 3' ' ' y - a h a i diem S thoa man la S ( 3 ; l ; - 2 ) , s f - l l - i ^ ! Khi do AD(x - 5; y - 3; z +1), BD(x - 2; y - 3;z + 4),CD(x - 1 ; y - 2; z) U'3' 3 ' • Liim nang luyfn ini VH ninn nyt. - jTjjwyt.). - x„m Bai 3.1.7. Trong khong gian O x y z , cho hinh hop chu nhat A B C D . A ' B ' C ' D ' C t y TNHH MTV DWH Khang Vift 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 J) Xac dinh tpa dp cac dinh ciia hinh chop do tinh khoang each giua hai duang th^ng cheo nhau A ' C va BD . Huong dan giai 2) Chung minh rang ASCD vuong, tinh di?n tich tam giac SCD va tinh c6 sin ciia goc hop boi hai mat phMng (SAB) va (SCD). 1) Taco A(0;0;0),B(1;0;0), D(0;2;0), A'(0;0;3). H i n h chieu ciia C len (Oxy) la C, hinh chie'u ciia C len Oz la A nen C(l;2;0). Hinh chieu ciia B',C',D' len mp(Oxy)va true Oz Ian lug^ la cac diein B,C,D va A ' nen B'(l;0;3), C'(l;2;3), D'(0;2;3). 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 . 4) Tren cac canh SA,SB,BC,CD Ian lupt lay cac diem M , N , P , Q thoa SM = M A , SN = 2NB, BP = 3PC, CQ = 4QD . Chung minh rang M , N , P, Q 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) Theo gia thiet cua debai, ta c6: N ( 1 - y;2y;0) MN.A'C = 0 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) Nen (*) <=> Do do M l _ x - y + 4y-4x-9x + 9 = 0 24 61 (17 88 44 -3x + 5y = l H + x + y + 4y-4x = 0 (53 106 6 1 ' 61 [-14x + 3y = -10 l)Ta ' 6?'61'" Vi M N la duong vuong goc chung cua hai duong t h i n g A ' C , B D nen d ( A • C, BD) = M N = ^(1 - X - y ) ' + (2y - 2xf + (3x " 3)' = ^ • CO A(0;0;0),B(a;0;0),D(0;2a;0),C(a;a;0).Dat SA = x S(0;0;x) = (-a; 2a; O), SC = (a; a; - x ) ^ DB = a^/5, SC = > / ? T 2 ^ B D . S C = a^ cos a = C O S ( S C , B D ) SCBD ^ + 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 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 c (SAB) la tam giac SAB ^a.2a nen ta suy ra cosP = ^ASAB _ 2 ^ASCD a^ \/3 Khi do 2) Cho a + b = 4 . Tim max V^.guj^ . \[3 Huong dan giai 1) Ta c6: C(a;a;0), B'(a;0;b), C'(a;a;b), D'(0;a;b) : ^ M ( a ; a ; | ) Suyra A ^ = (a;0;-b), A ^ = (0;a;-b), A ^ = (x-a)2+y2 + z2=x2+y2+(z-2a)2 =IS^^< ( x - a ) 2 + ( y - a ) 2 + z 2 = x 2 + y 2 IC^ 1) Tinh the tich ciia khoi t u d i f n B D A ' M . 1 3) Ta CO E(0;a;0). Gpi l ( x ; y ; z ) la tam mat cau ngoai tiep hinh chop SBCE IB^ =IS^ ^{z-2af Vay X= — 2 - x - y + 2z = a o a a i^2^^ ^ 2 z=a -2y + 4z = 3a . 4) Taco M(0;0;a) 2) Do a, b > 0 nen ap dung BDT Co si ta c6: 2 a a — V2y Dang thuc xay ra I 4 Vay max J M N A M P 3' ' 3 3 3a , MP = a ; - ; - a 3 64 27 2 a 9a Xac djnh toa dp cac dinh con lai va h'nh the tich khoi t u d i f n do. ' Huong dan giai (MNAMP).MQ = V^,NPQ=^(MNAMP).MQ — Taco: A G = ( - 2 ; 2 ; - l ) , A M - ( l ; 5 ; y ) S u y ra riABC = A G A 2 A M = (-12; - 2 4 ; nen M , N , P, Q khong dong phang. Vay v.. A'BDM b= l G ( l ; l ; l ) la trpng tam tam giac ABC. Duong thSng BC di qua M ^ 2 2 2^ = a a a 4 3 4 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 5 Suy ra M N = o 2j 4 27 a =a+b= 3—• 3 V4 2 8 + a2 = a , BP = - B C ^ P a;—;0 Do SN = - S B = > N VA'MBD = 3a^h a'b 4 = a + b = l a + l a + b>33/ia2b^a2b<-^ N2 Ban kinh R = IE = A ' B , A ' D = (ab;ab;a2) = > A ' M . A ' B , A ' D nen a -2x + 4z = 3a b a;a;-- x^ + (y - af + z^ = x^ + y^ + (z - 2af lE^ =IS^ Trong khong gian vdi h? tpa dp Oxyz cho hinh hpp chu n h l t a 40' -24) Phuong trinh (ABC) : x + 2 y + 2 z - 5 = 0. I Ug^ A G A HABC = ( 6 ; 3 ; - 6 ) 7^ 4;4;-j - t y 1 i\tltl MIV DWH § 2. LAP x = 4 + 2t PHLTONG TRlNH M A T KhangVift PHANG 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 ) : 2t + l;t + 5 ; - 2 t - Y Ta c6: B 4 + 2t;4 + t ; - - - 2 t AB = Nen ta dugc phuong trinh : ( 2 t + lf+(t Ot2+4t + i ^= 0 o t 4 Suy ra B = , Neu hai vec to a,b khong ciing phuong va c6 gia song song hoac n i m tren 11,2 + 5)2+(2t + Y) - 2 7 (a) thi a A b = n la VTPT cua (a). , Neu mat ph5ng (a) di qua ba diem phan b i f t k h o n g thang hang A,B,C thi AB A AC = n la VTPT cua (a). 73 -2±- Neu (a)//(p) thi n^ = np . 4-N/3 1 + 2N/3' f ^ . 4 + ^/3.1-2^/3 , C 2 , Neu A 1 (a) thi n^^ = u ^ . 2 . x - l +t Neu (a)//(P) thi i^//(a). . 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 z = l + 2t m Suy ra D ( l + t ; l + 2 t ; l + 2t) Cach 2: Gia su phuong trinh (a) c6 dang: Ma GD = N / D A ^ ^ ^ A G ^ = ^ = 3N/2 =^ 3 t =3^f2^i = ±^f2 3 D o d o S(1 + N/2;1 + 2>/2;1 + 2V2) hoac S(l - N/2;1 - 2 ^ 2 ; ! - 2^/2). The t.chcuatudi?n: V = 1DG.SABC -^^^ =— 3 3 4 ( 0. 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: mx + y + nz + p = 0. Chu y: Neu mat phMng (a) d i qua M(Xo;yo;Zo) t h i p h u o n g trinh cua (a) c6 dang: a ( x - x ^ ) + b ( y - y , , ) + c ( z - Z Q ) = 0. Cac dang lap phuong trinh mat phSng. Dang 1: Lap phuong trinh mat phdng biet mat phdng di qua ba diem A, B, C. CO n = AB A AC la VTPT va A la diem di qua. ' 3.2.1. Lap phuong trinh mat phAng (a) bie't ( a ) B(0;-l;2), C ( - 2 ; 3 ; - l ) ^^S^c hinh chieu cua A len cac true tpa dp. „ , d i qua Clin lui/iii Ihi nil Ilnili UOL Cty TNHH MTV DWH Khang \^uyenTafT}iH Led gidi. 1) Taco AB(-l;-2;l), AC = (-3;2;-2) AB A AC = (2;-5;-8) Suy ra n„ = (2; -5;-8) . Do do, phuang trinh ciia ( a ) la: ^„ 2 ( x - l ) - 5 ( y - l ) - 8 ( z - l ) = 0 hay 2 x - 5 y - 8 z +11 = 0. 2) Goi Aj, A j , A3 laa lugt la hinh chie'u ciia A len cac true Ox, Oy, Oz. Ta CO Aj(l;0;0), A2(0;l;0), A 3 ( 0 ; 0 ; ; ) . Vi (a) di qua Aj,A,,A3 I nen phuang trinh c6 dang: x y z - + — + — = 1 hay x + y + z - l = 0. I l l ^ ^ Dang 2. Lap phuang trinh mat phang (a)song song v&i mat phang Cp; hoac (a) 1 A B . Ta CO n^^ hoac AB la VTPT cua ( a ) . Vi 3.2.2.((3): Lapaxphuong phSng biet: Chii dv y: Neu + by + cztrinh + d mat = 0 thi np -(a) (a;b;c). 1) (a) diqua M(2;3;1) vasong song voim^t phing (P):2x + 3 y - z + l = 0 2) (a) la mat ph^ng trung true cua do^n A B vol A(1;4;2), B(-3;2;-1) . Lai gidi. ^ 1) Taco r^ = (2;3;-l). Vi (a)//(p) nen n ^ = njj = (2; 3;-l) .Phuang trinh mat phing (a) la: 2 ( x - 2 ) + 3 ( y - 3 ) - l ( z - l ) = 0 hay 2x + 3 y - z - 1 2 = 0 . 2) Vi (a) la mat ph^ng trung tryc cua doan AB nen AB = (-4;-2;-3) la VTPT ciia ( a ) . Phuong trinh mat phang (a) la: 4 ( x - l ) + 2 ( y - 4 ) + 3 ( z - 2 ) = 0 hay 4x + 2y + 3 z - 1 8 = 0. Dang 3: Lap phifdng trinh mat phang (a) biet (a) song song vdi hai vect(< khong cung phi^dng a,b Khi do n„ = a A b . Vi$t Chii y: , Neu (P) chua (hoac song song) vol A B thi gia cua vec to A B se nam tren (hoac song song) voi (P). , Neu (P) -L (Q) thi VTPT cua mat phSng nay se c6 gia nam tren hoac song song voi mat phSng kia. Vi dM 3.2.3. Viet phuong trinh mat phang (a) biet: 1) (a) chua A(l;l;2), B(1;-1;1) va song song voi CD trong do C(0; 3; 0), D(-2;0;-l). 2) (a) chua A , B va vuong goc voi mat phSng (p):x + y + z + l = 0 3) (a) di qua C va vuong goc voi hai mat phMng (P) va (y): 2x - y + z - 3 = 0 . 4) (a) di qua A, B va each deu hai diem M(3;-5;-1), N(-5;-1;7) . Lai gidi. 1) Taco A B = (0;-2;-l), CD = (-2;-3;-l). Vi (a) di qua A, B va song song voi CD nen n^ = A B A CD = (-1;2; -4). Phuang trinh (a): x - 2y + 4z - 7 = 0. 2) Taco n p = ( l ; l ; l ) . Vi (a) chua A , B va song song voi (p) nen n^ = ABAnp ==(-l;-l;2). Phuong trinh mat phSng ( a ) : x + y - 2z + 2 = 0 . 3) Taco n^^ = ( 2 ; - l ; l ) . ,. Vi mat phSng (a) vuong goc voi hai mat phSng (P) va (y) nen n„ = np A n^ = (2;l;-3). Phuang trinh mat phSng ( a ) : 2x + y - 3z - 3 = 0 . Vi (a) each deu hai diem M, N nen ta c6 cac truang hg-p sau T H 1: M N / / ( a ) . K h i d 6 n^ = A B A M N = (-12;8;-16). Phuong trinh ( a ) : 3x - 2y + 4z - 9 = 0 . T H 2: M N c3t (a) tai I, suy ra I la trung diem doan M N . D o d o I(-1;-3;3). Khi do n„' = A1 A BI = (-6; 2; -4). f huong trinh ( a ) : 3x - y + 2z - 6 = 0 . c r y i j V f j H MTVnWHKhans: Dang 4. Viet phuang trinh mat phdng di qua m6t diem vd lien quan den hhodng cdch hoac goc Phuong trinh di qua M(xQ;yo;zQ) c6 dang a(x-XQ) + b ( y - y g ) + c ( z - Z o ) = 0 vai + >0. Dua vdo dieu ki$n khoang each hoac goc ta tim du b = -2a, ta chpn a = 1 => b = c = - 2 . phuong trinh mat p h i n g (a): x - 2y - 2z + 6 = 0 . 3 2 , a = — b = > b = — a , ta chon a = 3 => b =-2,c 2 3 3) Vi mat phSng (a) d i qua C ( l ; 3 ; - 2 ) nen phuang trinh c6 dang a ( x - l ) + b ( y - 3 ) + c(z + 2) = 0<=>ax + by + c z - a - 3 b + 2c = 0. (a) diqua D ( - l ; 0 ; 3 ) nen -a + 3 c - a - 3b + 2c = 0 a=-^^^ . Mat khac d ( M , ( a ) ) = d ( N , ( a ) ) nen ta c6 (P) bang 5. 2) (a) d i q u a A ( 0 ; 1 ; 2 ) , B(2;1;3) va each O mot khoang bang 2 3) (a) =-6. phuang trinh mat phang (a): 3x - 2y - 6z + 1 4 = 0 . V i dv 3.2.4. Viet phuang trinh mat phang (a) biet 1) (a) song song voi ( Q ) : 2x - 3y - 6 z - 1 4 = 0 va khoang each tir O den |-a - b - e - a - 3b + 2c = 3a + 2b + 2c - a - 3b + 2c di qua hai diem C ( 1 ; 3 ; - 2 ) , D ( - 1 ; 0 ; 3 ) va each deu hai diem <=> M ( - l ; - l ; - l ) va N(3;2;2) I — 2a + 4 b - c = -2a + b - 4 c -3b-3c-r = c, ta chpn c = l = > b = l,a = l . Phuang trinh mat phSng ( a ) la: Lai gidi. Mat khac d ( 0 , ( a ) ) = 5 = > - ^ = 5 ^ m = ±35. Phuang trinh ( a ) : 2x - 3y - 6z ± 35 = 0 . 2) Vi mat phang (a) d i q u a A(0;1;2) nen phuang trinh C6 dang ax + b ( y - 1 ) + c(z - 2) 0 <=> ax + by + cz - b - 2c = 0 . ( a ) d i q u a B(2;1;3) nen 2a + b + 3 c - b - 2 c = 0 O C = - 2 a . Mat khac d ( 0 , ( a ) ) = 2 nen ta c6 =2o a= I- 1 3N/3 ' 1) Vi (a)//(Q) nen phuang trinh (a) codang: 2 x - 3 y - 6 z + m = 0 . b + 2c •b = c 2a + 4 b - c = 2 a - b + 4c - 2 a - 4 b + c| = |2a-b + 4c 4) (a) d i qua E ( 0 ; - 1 ; 1 ) , F(2;1;2) va tao voi mp ( p ) : x + y + z - 1 = 0 mot goc (p thoa cos(p = Vu-I b-4a =2N/5a^+b^ x+y+z-2=0. K -3b-3c , 5c-3b -3c-3b , 1 3 15 Hke: , ta CO = <=> b = —e,a = —^c. —-— H ' a chon c = -12 => a = 45,b = -52 . ^ P i u o n g trinh mat phSng ( a ) : 45x - 52y - 12z + 81 = 0. 4) Vi mat phSng (a) d i qua E ( 0 ; - l ; l ) nen phuong trinh c6 dang ax + b ( y +1) + c(z - 1 ) = 0 <:i> ax + by + cz + b - c = 0. ^a) d i q u a F(2;1;2) nen 2a + b + 2c + b - c = 0 < » c = - 2 a - 2 b . ^atkhac |cos(n„,np) = COS(p = a+b+c 3^3 VsS^Tb^T?) 3V3 ^ 3 a + b =^a^+b^+(2a + 2b)^ o 9 ( a + b)^ =5a^ +8ab + 5b^ "=^23^+5ab + 2b^ =0<=>a = - 2 b , a - - i b . • 3 = -2b=>c = 2b,chQn b = - 1 =>a = 2,c = - 2 . » (b - 4a)^ = 4 (Sa^ + b ^ ) « 4a^ + Sab + Sb^ = 0 » ^huong trinh (a) la: 2x - y - 2z + 1 = 0. I , Dang a = - - b , t a c h Q n b = - 2 = > a = l , c = 2. 2 Phuang trinh (a) trinh doan chdn 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 la: x - 2 y + 2 z - 4 = 0 . Dang 5: Viet phuong 6: Phuong trinh mat phdng di qua giao tuyen hai mdt phdng ciia t r i n h (a) c6 d a n g - + f + - = 1 . a b c '^jid^ ' -, I.' .... :i 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 (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 Xet he p h u a n g t r m h \ u i I a 2 X + b j y + C2Z + d 2 = 0 [aiX 3) O A = OB = O C . 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: (a,): X + y + z - 3 = 0; ( t t j ) : 2x + 3y + 4z - 1 = 0 . 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 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 ) . Lai giai. 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)V I n j ?t kn2 nen hai mat phang (QJ) Xet h ^ p h u a n g t r i n h Cho z = - 1 , ta C O he • Cho z = - 2 , ta C O h? VI (a) va ( a 2 ) cat n h a u . Diem M la true t a m t a m giac A B C k h i va chi k h i Me BM.CA = 0 2x + 3 y + 4 z - l = 0 ' x+ y = 4 x-7 2x + 3y = 5 ' y = -3^ .M(7;-3;-l)e(ai)n(a2) ''^^^^N(6;-l;-2)e(ai)n(a2)- 2x + 3y = 9 d i qua giao t u y e n cua h a i m§t ph4ng {a^) M,N. v a ( t t z ) n e n ( a ) d i qu^ .--'^ 1 9 (a) AM.BC = 0 x + y + z-3::=0 x+y = 5 X V z Phuang t r i n h m a t p h a n g (a) c6 d a n g — + — + — = 1. a b c 1 9 4 Mat phang (a) d i qua d i e m M ( l ; 9 ; 4 ) nen - + — + - = 1 (1). 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); 4 , - + - + - = 1 o a b c 9b = 4c a = 4c ^ a = 98;b = 28;e = i ^ 9 2 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. -1 ^achl: Taco: d(0,(a)) = a^ b^ • c2 Va^ h'2 c„2 Ta C O A M - (6; - 3 ; - 2 ) , A M = (5; - 1 ; - 2 ) ^ A M A A N = (4; 2; 2 ) . Bai toan t r o thanh, t i m gia trj nho nhat ciia T = 2 - + A + ^ 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 ) ^ ' b , c ; ^ 0 t h o a m a n ! + - + ! = 1 (]).' a b c la: 2x + y + z - 3 = 0 . "^P d u n g b d t Bunhiacopski ta c6: , v d i cac so t h u c 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 Nen suy ra T > ^ . Dau dang thuc xay ra khi 1:1 = 9 : 1 = 4:1 • ^ 1 9 a ^ 4 , b ' >• • ^ o a = 9b = 4c = 98. c ^ 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). Vi mat phang (a) luon d i qua diem co'djnh M nen a=5 i i + - ^ - ^ = l « a 2 + 2 a - 3 5 = 0«> 2a - 4 2a a = -7 40 y j a > 2 nen a = 5=>b = 2;c = , phuang trinh m|t ph3ng can t i m la (a): 8x + 20y - 37z - 40 = 0. , Meu b < 0 = > c = - ^ a , b = - l ^ ^ , a > 2 1, a |I = , « 4-2a nentu ( l ) t a c 6 , 29a - 35 . 0 « a = 2a 2 Vi a > 2 nen khong c6 gia tri thoa man. d(0,(a)) = O H < O M = >/98. 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 « . . Diem M(xQ;yo;zo)€(a):ax + by + cz + d = O o a x o + byo + cZo + d = 0. X + 9y + 4z - 98 = 0. 3) Vi OA = OB = OC nen a = b = c , do do xay ra bo'n truong h^p sau: • Truong hop 1: a = b = c. Tu (1) (1) suy suy ra 1+ + -— +1 = 1 Tu ra 1 aa aa a a = 14, nen phuang trinh (a) la: Truang hap 2: a = b = -c. T u (1) suy ra 1 + ^ - 1 = 1 « a = 6, nen phuong j'ong hop 2: a = 1 trinh (a) la x + y - z - 6 = 0. - • Truong hgp 3: a = - b = c. T u (1) suy ra 1 9 Si Si 4 + - = l o a = -4, nen phuang Si trinh (a) la x - y + z + 4 = 0. • Truong hp-p 4: a = - b = -c. T u (1) c6 1 9 Si 4 Si Si = l<=>a = -12, nen phuang trinh (a) la x - y - z +12 = 0. Vay CO bon mgt phang thoa man la x + y + z - ; 4 = 0, va cac mat phang x + y - z - 6 = 0 , x - y + z + 4 = 0, x - y - z + 12 = U. 4) V i x^ > 0,Z(-- < 0 nen G (P) 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„ A(3;3;3),B(1;2;4),C(0;3;2) vam|tphSng ( a ) : x + y + z - 3 = 0. 1) Tim tpa dp hinh chie'u cua A len (a). x + y + z - 1 4 = 0. . H a > 0,c < 0, do do 8 0 A = 120B + 16 = 370C o 8a = 12 b +16 = -37c. j ) J i m tpa dp diem M thupc (a) sao cho M A + MB nho n h a t Laigidi. '"^9i H la hinh chieu ciia A len (a) . x = 3+t Phuang trinh A H : y = 3 + t = > H ( 3 + t;3 + t;3 + t ) . z=3+t ^» H e ( a ) nentaco: 3 +1+ 3 +1 + 3 + 1 - 3 = 0 o t = - 2 H(1;1;1) . ^' ^oi mSi diem K ( x ; y ; z ) ta d?t f(K) = x + y + z - 3 . CO f(A) = 6, f(B) = 4 => A , B nSm cung phia so voi m^t ph5ng ( a ) . A ' la diem dol xung voi A qua ( a ) , suy ra A ' ( - 2 ; - 2 ; - 2 ) 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 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 Ma M 6 ( a ) nen -2 + 3 t - 2 + 4 t - 2 + 6 t - 3 = 0=>t = —=>M sau. THl:AO//(a) x = -2 + 3t Phuang trinh (a): 7x - y - 6z + 3 = 0. n J_.10.28^ Vi d\ 3.2,8. Viet phuang trinh mat ph5ng (a) biet: T H 2: (a) n OA = { l ) , suy ra I la trung diem doan OA => I 1 1 K2'2'2) (3 3 5^ rs 3 ^ Suy ra I B = l2'~2'2j = > B C A I B = [2 2 ) la VTPTcua (a). 1) (a) di qua hai diem A(1;1;1),B(2;-1;3) va song song voi OC v6i Ph Phuang trinh (a): x + y -1 = o . U3'13'13 3) Tonghgfp "vTd Vi dv 3.2.9. Lap phuang trinh mat phing (P), biet: C(-l;2;-l). 2) (a) di qua M ( l ; l ; l ) , vuong goc voi (P): 2x - y + z - 1 = 0 va song song vol • 2 1 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;—\ \j -3 • 3) (a) vuong goc vai hai mat ph3ng (P):x + y + z - l = 0 , (Q):2x-y + 3z-4 = 0 ^ 7 den (P) bang—•== . 6V3 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. Suy ra Suy ra phuang trinh (a): 2x + y - 3 = 0 . 2a-b + d = 0 Jb = 2a + d 5a + 'b + c + d• = 0 ' ^ l c = - 7 a - 2 d ' 2) Taco: n,^ = (2;-l;l), u^=(2;l;-3) l|c + d Do [(a)l(P) (a)//A J^ltkhac: d(M,(P)) = - ^ = ^ _ "a ="p ^ " A =(2;8;4). 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)^ I' n = n ^ A n ^ = (4;-l;-3) lamVTPT. Suy ra phuang trinh (a) c6 dang : 4 x - y - 3 z + d = 0. * V a.-d - ^ 2 7 a 2 + 3 2 a d + 5d2=o<^ 27 Meit khac: d(0,(a)) = %/26 nen ta c6: = V26 => d = ±26. N/26 Vay phuang trinh (a): 4x - y - 3z ± 26 = 0. d ~ -a b - a;c = -5a . Suy ra phuang trinh (P) la: ^^ + a y - 5 a z - a = 0 o x + y - 5 z - l = 0. Cty TNHH MTV DWH d = - — a => b = * yf a;c = - — a. Suy ra p h u o n g t r i n h (P) la: 5 5 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 x - l _ y +l _ z - l 5 x - 1 7 y - 3 6 z - 2 7 = 0. x+l _ y - l _ x = -2t z va d3 : y = - i - 4 t . 2) G i a s u ( P ) : a x + b y + cz + d = 0 a = a+2b+c+d=0 ViA,Be(P)=^ _2^^^,^3^^d = 0' 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 . 3 5b + 5c 5b-c sao cho A B = \/l3 . 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) = 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 = • V o i c = - b , ta chon b = - 1 => c = 1 =>a = l , d = 0 5 • V o i c = 3 b , ta chon b = 1 => c = 3,a = — , d = 3 P h u o n g t r i n h ( a ) : 5x + 3y + 9z - 20 = 0 . 20 3 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 . _ x - 1 _ y + 1 _ z + 2 _^ , x-2 _ y + 2 _ z-1 • 2 -4 • ~ -1 ' ^' 1 ~ 2 d j C O U2 = ( 2 ; 3 ; - l ) la VTCP vahaiduong ^^^7^^^^;^;^^;^—;^^ 1 " 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). Phuong t r i n h ( a ) : x - y - z - 1 = 0 . ; 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. Ta c6: D u o n g t h i n g d i qua M ( l ; - l ; - 2 ) , VTCP u ^ = ( 2 ; l ; - l ) 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 ) D o (P) d i qua A va d^ nen np = A M A Uj = ( - 8 ; 1 0 ; - 6 ) Suy ra p h u o n g t r i n h ( P ) : 4x - 5y + 3z - 3 = 0 . 2) Xet h$ p h u o n g t r i n h 2t-t' = l _ l + t = - 2 + 2 t ' ^ t - 2 t ' = -l<=>t = t' = l _2-t = l-4t' Voi x = l=> AB = (0;2;-3), A(-l;l;0)ed2 Suy ra n = A B , u ta c6 u=(2;3;-l) la VTCP cua dj A€(a). = (7;-6;-4) laVTPTcua (P). Phuong t r i n h (P): 7x - 6y - 4z +13 = 0 . 1) T a c o : A M = ( 0 ; - 3 ; - 5 ) l + 2t = 2 + t ' — . Vl45 Lai gidi. Phuong trinh (a): x - y + z = 0 . ~ ' 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 M a t k h a c : d ( C , ( P ) ) = 2 d ( D , ( P ) ) » | 2 a - b + c + d| = 2|3b + c + d o ".H, z = - l + 2t - b + 2c d = - Khang Vift - t + 4t' = 3 • V6ix = i=>XB = (-^;-^;-^). 3 3 3 3 S u y r a ri = r - 3 A B , u 1 = (-14;11;5) l a V T P T c u a i. •„ (P). Phuong t r i n h ( p ) : 14x - l l y - 5z - 25 = 0 . ^) 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 ) . ' va