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T R U N G T A M LUYCN T H I D A I HOC V I N H VI£N SAI G O N Tdng chu bi§n: PHAM H 6 N G D A N H NGUYEN PHU KHANH - N G U Y I N TAT THU NGUYEN TAN SIENG - TRAN VAN TOAN - NGUYEN ANH TRUCfNG (Nhdm giao vien chuyen luyen thi B^i hpc) PHUONG PHAP GIAI TOAN HtNH HOC • theo chuyen de H I N H HOC T R O N G K H O N G G I A N * H I N H HOC T Q A OO T R O N G K H O N G G I A N H I N H HOC T O A OO T R O N G M A T P H A N G THU VIEN l\m 8 I N H THUAN] N H A X U A T B A N D A I HOC QU6c GIAHA NQI " Ctij TNHH N H f l X U R T B R N D f l l H O C Q U O C G l f l Ht{ MTV DVVH Khang Viet NOI 16 Hang Chuoi - Hai Ba TrUng - Ha Npi Dien thoai : Bien t a p - Che ban: (04) 39714896; Hanh chinh: (04) 39714899; Tona bien t a p : (04) 39714897 Fax: (04) 39714899 P H U O N G P H A P T O A D O T R O N G IVIAT P H A N G A , LY THUYET G I A O K H O A I. Tpa dp trong mat phang. Chiu trdch nhiem xuat ban Gidm doc - Tong bi&n tap : • Cho u ( x p y j ) ; v(x2;y2) va k e R . K h i do: 1) u + v = (xi + X 2 ; y i + y 2 ) TS. P H A M T H j T R A M 3) k u = ( k x i ; k y i ) Bien tap : N G Q C LAM Che : C O N G TY KHANG V I E T : C O N G TY KHANG V I E T ban Trinh bay bia 4) "''""''^ 2) u - v = ( x i - X 2 ; y i - y 2 ) Z=Jx\+y\) u=vc^r^ 6) U . V = X ] X 2 + y ] y 2 = > u l v < ; : > u . v = 0<=> \-^\2 + y ] y 2 = 0 • Haivecta u ( x j , y j ) ; v ( x 2 ; y 2 ) c i i n g p h i r a n g v a i n h a u <=> • Goc g i i j a hai vec to u ( x j , y j ) ; v ( x 2 ; y 2 ) : Tong phdt hanh va doi tdc lien ket xuat ban: cos(u,v)= U.V u M^^k CONG TYTNHH lpiS|r MTV 1) A B = ( x B - X A ; y B - y A ) 2) ^^=^3 = ^{x^ - x + {y ^ - y _ X A + X B I ~ 3) Email: khangvietbbokstore@yahoo.com.vn Website: www.nhasachkhangvlet.vn V Cho A ( x ^ ; y ^ ) ; B ( x B ; y B ) . K h i do : DjCH Vg VAN HOA KHANG V I E T f D i a c h I 71 Dinh Tien Hoang - P Da Kao - Q 1 - TP HCN/I ^ Dien thoai: 0873911569^^ 39105797 - 39111969 - 39111968 Fax: 08. 3911 0880 ^ XiX2+yiy2 t r o n g d o I la t r u n g d i e m ciia A B . y • AB 1 CD o AB.CD - 0 • Cho tarn giac A B C v o i A{x^;y^), B(xB;yB), C{x^;y^). K h i d o t r o n g tarn SACK L I E N K E T PHLfONG PHAP GIAI TOAN M a so: HINH H Q C THEO CHUYEN DE G ( x ( , ; y g ) ciia tarn giac A B C la : 1L-321DH2012 In 2.000 c u o n , kho 1 6 x 2 4 c m T a i : Cty T N H H MTV IN A N MAI T H j N H DLfC V _ X A + X B + X C X G ^ yG= I I I . PhirotTg trinh duong thang Dja chi: 7 1 , Klia Van Can, P. H i e p Binh C h a n h , Q . Thu Dufc, TP. Ho Chi M i n h 1. 'Phuang trinh duong thdng So xuat bSn: 1335 - 201 2/CXB/07 - 21 5 / D H Q G H N ngay 0 6 / 1 1 / 2 0 1 2 . 1.1. Vec to chi phucmg (VTCP), vec to phdp tuyen (VTPT) cua duong thang: Cho d u o n g t h a n g d . Quyet d i n h xuat b5n so: 3 1 8 L K - T N / Q D - N X B D H Q G H N , cap ngay 12/11/2012 In xong va nop luu chieu Q u y I n a m 201 3 ,, ,^, • n = (a;b) ?t 0 g o i la vec t o p h a p t u y e n cua d neu gia ciia no v u o n g v o i d . 3 Phiam^ phiip giui loiin llinli hoc Iheo chuycn de- Nguyen Phti Khdnh, Nguyen Tat Thti • Cty TNHH MTV DWH Khang Viet u = ( u j ; u 2 ) ^ 0 goi la vec ta chi phuong cua d ne'u gia cua no trung hoac d(M,(A)): song song voi duong thang d. Mot duong thang c6 v6 so VTPT va v6 so VTCP ( Cac vec to nay luon cung phuong voi nhau) axp + byp + c Va^+b^ 5. (phuong trinh duang phdn gidc cua goc tao boi hai duang thdng Cho hai duong thang d^ : a^x + b^y + c^ = 0 va d2 : a j X + b2y + Cj = 0 • Moi quan he giua VTPT va VTCP: n.u = 0 . Phuong trinh phan giac ciia goc tao boi hai duong thang la: - , v , • • Ne'u n = (a; b) la mpt VTPT cua duong thang d thi u = (b; -a) la mot VTCP ajX cua duong thang d . • ~!( 5 + b^y + Cj f + • Duong thang AB c6 AB la VTCP. III. Phuang trinh duong tron. 1.2. Phuwig trinh dumig thang 1.2.1. Phuatig trinh tong qudt cua duong thang: 1. d(I, A) = R ' [a2X + b 2 y + C2 =0 • Neu (I) v6 nghiem thi d^ / /d2 . • Ne'u (I) v6 so nghiem thi d^ = d j • Ne'u (I) CO nghiem duy nha't thi dj va d2 cat nhau va nghiem ciia he la toa do giao diem. ' 3. Goc giua hai dijcang thdng. Cho hai duong thang dj : a j X + b^y+ Cj =0; d2 :a2X + b2y + C2 = 0 . Goi a la goc nhon tao boi hai duong thang dj va d2 . Ta CO : cosa = aja2 + bjb2 ^/a^Tb^ ^/af+b 4. JChodng each tit mot diem den ducrng thdng. Cho duong th5ng A : ax + by + c = 0 va diem M ( X Q ; y ^ ) . Khi do khoang each tu M den A dugc tinh boi cong thuc: • Duong tron (C): (x -a)^ + (y - b)^ = R^ c6 hai tiep tuyen cung phuong voi Oy la x = a ± R . Ngoai hai tiep tuyen nay cac tiep tuyen con lai deu c6 dang: y = kx + m . IV. E lip 1. 'i)inh nghra.-Trong mat phang cho hai diem co'djnh Fi,F2 c6 Y^Yj =2c. Tap hop cac diem M cua mat phang sao cho MF^ +MF2 =2a (2a khong doi va a > c> 0) la mot duong elip. • F,,F2 : la hai tieu diem va 2c la tieu cu ciia elip. • MF|,MF2 : la cac ban kinh qua tieu. 2. Phuang trinh chinh tdc cua elip: 4 +4 a 2 b^ Vay diem M(xo;y(,) e (E) • = ^ voi b^=a^-c^. = 1 va d hlnh dang cua elip: Cho (E): — + ^ a b • True doi xung Ox,Oy . Tarn do'i xiing O . = 1, a > b . +) MF^ = ex„ + a va MF2 = e X ( , - a khi +) MFj = -exp - a va MF2 = -exp + a khi j, • Dinh: A[(-a;0), A2(a;0), 6^(0;-b) va 62(0; b ) . A^A2 = 2a goi la do dai true Ion, B]B2 = 2b goi la do dai true be. ^ Q • Tam sai: e = — = a ..2 .2 0. XQ < 0. 2 M ( x o ; y o ) 6 ( H ) : \ - J ^ = l « ^ - f j = l vataluonco X(,]>a. D a • Noi tiep trong hinh ehir nhat co so PQRS CO ki'ch thuoc 2a va 2b voi b^ = a^ - e^. • 2 . • Tieu diem: F|(-c;0), F2(e;0). XQ > a b VI. Parabol j.<^inhnghia: Parabol la tap hop cae diem M cua mat phang each deu mot duong thang A co'dinhvamot diem F co dinh khong thuoe A . 0 <1 A : duong chuan; F : tieu diem va d(F,A) = p > 0 la tham so'tieu. a s a a^ Hai duong chuan: X = ± — = ± — e e Bi R 2. 'Phuxmg trinh chinh tdc cua ^arabd: = 2px 3.jrinh dang cua Parabol ( a > 0 ) la mpt Hypebol. • Fp F2 : la 2 tieu diem va F|F2 = 2e la tieu eu. M ( x ; y ) e ( P ) : MF = x + ^ voi x > 0 . B, CAC BAI THlfONG GAP • 1VIF[,MF2 : la eac ban kinh qua tieu. § 1. x^ 2. 'Phimng trinh chinh idc cua hypebok a^ y^ = 1 voi h^=c^-a^. 3. Tinh chat vd hlnh dang cua hypebol (fi): cAc BAI T O A N C O B A N 1. Xg.p phuang trinh duang thang. De lap phuong trinh duong thang A ta thuong dung cac each sau • True doi xung Ox (true thuc), Oy (true ao). Tam doi xung O . • Tim diemM(xo;yo) ma A di qua va mot VTPT n = (a;b). Khi do phuong • Dinh: Aj(-a;0), A2 (a;0). D Q dai true thuc: 2a va do dai true ao: 2b. trinh duong thang can lap la: a(x - XQ) + b ( y - yp) = 0 . • Tieu diem Fi(-e; 0), Fj ( c; O) . • Gia su duong thang can lap A : ax + by + e = 0 . Dua vao dieu kien bai toan ta tim dugc a = mb,c = nb. Khi do phuong trinh A : m x + y + n = 0. Phuong phap • Hai tiem can: y = ± —x a • Hinh eho nhat co so PQRS c6 kieh thuoe 2a, 2b voi b^ = c^ - a^. nay ta thuong ap dung doi voi bai toan lien quan den khoang each va goe • Phuong phap quy tich: M(xQ;yQ)e A:ax + by + e=^Oc:> axy + by^ + e = 0 . Vidu 1.1.1.Trong mat phSng voi he toa do Oxy cho duong tron • Tam sai: e = — = a (C):(x-])2+(y-2)2=25. 1) Viet phuong trinh tiep tuyen ciia (C) tai diem M(4;6), ' • Hai duong chuan: x = ±— = ± — 2) Viet phuong trinh tiep tuyen cua (C) xua't phat tu diem N ( - 6 ; l ) Cty TNHH MTV DWH Phucntg phap giai ToAn Ilinh hoc theo chuycn lic- Nguyen Pliii Khanh, Nguyen Tat Thii Khang Viet D u a vao gia thie't cua bai toan ta t i m dugc a , b , c . Cach nay ta t h u o n g ap 3) T u E(-6;3) ve hai tie'p tuye'n EA, EB (A, B la tie'p diem) den (C). Viet d u n g k h i yeu cau viet p h u o n g t r i n h d u o n g tron d i qua ba d i e m . p h u o n g t r i n h d u o n g thang A B . Vi du 1.1.2. Lap p h u o n g t r i n h d u o n g tron (C), bie't 1) (C) d i qua A ( 3 ; 4 ) va cac h i n h chie'u ciia A len cac true toa do. D u o n g tron (C) c6 tam 1(1; 2 ) , ban k i n h R = 5 . 2 1) Tie'p tuyen d i qua M va v u o n g goc v o i I M nen nhan I M = (3;4) l a m VTPT N e n p h u o n g t r i n h tie'p tuye'n la: 3(x - 4) + 4(y - 6) = 0 <=> 3x + 4y - 36 = 0 . 2) Gp i A la tie'p tuye'n can t i m . A : a ( x + 6) + b ( y - l ) = 0<=>ax + by + 6 a - b = 0, a^ + b^ 7a+ b G i a s i i ( C ) : x ^ + y ^ - 2 a x - 2 b y + e = 0. •=5o 7a + b = o24a2+14ab-24b2 = 0 o 2 4 • • 4 a =—b 3 5 ^ o{7a + b)^ = 2 5 ( 3 ^ +b^) - 6 a - 8 b + e = -25 - + 1 2 - - 2 4 = 0c^ b a=-lb' 3 4 thay vao n ta c6: — b x + by - 9b = 0 «• 4x - 3y + 27 = 0 . 3 - 8 b + e = -16 [(a - l)(a + 6) + (b - 2)(b - 3) = 0 a^ + b^ - 2 a - 4 b - 2 0 = 0 a-b a-7b V2 5V2 4 =- <=>b = -2a,a = 2b thay vao (1) ta CO dugc: (a - if- + 4a^ = - <=> 5a^ - 4a + — = 0 p h u o n g t r i n h nay v 6 n g h i e m 9 T u do ta suy ra duoc A e A : 7 x - y + 20 = 0. T u o n g t u ta cung c6 dug-c B e A = > A B = A = > A B : 7 x - y + 20 = 0 . 2. Cdch lap phimng trinh dizcrng tron. De lap p h u o n g t r i n h d u o n g t r o n (C) ta t h u o n g su d u n g cac each sau Cdch 7 ; T i m tam I(a;b) va ban k i n h ciia d u o n g t r o n . K h i do p h u o n g t r i n h . Cdch 2 ; G i a su p h u o n g t r i n h d u o n g tron co dang: x^ + y^ - 2ax - 2by + c = 0 8 7 +b a^ + b^ + 5 a - 5 b = 0 = ^ 7 a - b + 20 = 0 d u o n g tron co dang: (x - a ) ^ + ( y - b)^ = 7 Do (C) tie'p xuc v o i hai d u o n g t h i n g A ^ A j nen d ( I , A j ) = d ( I , A2) • b = -2a <=> e= 0 2) Goi I(a;b) la t a m ciia d u o n g t r o n (C), v i l € ( C i ) nen: ( a - 2 ) 3x + 4y +14 = 0 va 4x - 3y + 27 = 0. =25 3 a =— 2 <=> •!b = 2 . Vay p h u o n g t r i n h (C): x^ + y^ - 3x - 4y = 0 . Vay CO hai tie'p tuye'n thoa yeu cau bai toan la: 3) Goi A ( a ; b ) . T a c 6 : Ae(C) (a-1)^ + ( b - 2 ) ^ -6a + c = - 9 Do A , A p A 2 e ( C ) nen ta co he: a = ^b 4 3 7 thay vao (*) ta c6: - b x + by + - b = 0 o 3 x + 4 y + 14 = 0. lA.NA = 0 va tiep xiic v o i hai A,(3;0), A2(0;4). Va^ + b^ 3 a=-b =- 1) Goi A i , A2 Ian i u g t la h i n h chie'u ciia A len hai true Ox, O y , suy ra (*) Ta c6: R o 2) (C) CO tam n a m tren d u o n g t r o n ( C j ) : (x - 2)^ + y 4 d u o n g thc^ng A, : x - y = 0 va A2 : xXffigidi. - 7 y = 0. D o A d i qua N nen p h u o n g trinh c6 dang d(I,A) = 2 • 4 9 4 8 -^''i-' a = 2b thay v a o ( l ) t a c o : ( 2 b - 2 r + b ' ' = - < : : > b = - , a = - . o 0 0 Suy ra R = D ( I , A , ) = ( Vay p h u o n g t r i n h ( C ) : x I 3. Cac diem, ctqc biet trong tam Cho t a m giac A B C . K h i do: 8l 2 r 4^ ' — + y - 5 , 5j gidc. 8 25 -:l.:J...... (1) Phumig phdpgidi • Trong tam G 3 ' 7(x-l) + (y-3) = 0 j7x + y-10 = 0 Ma GH = -2GI. Suy ra I,G,H thang hang Ta CO4GH =4^ 32 "l6'l6 [) Goi K{x; y) la tam duong tron noi tiep tam giac ABC. Ta c6: KAB = KAC <=> KBC-KBA AK,AB) = (AK,AC COS(AK, A B ) = COS(AK, A C BK,BA = (BK,BC) cos (BK, B A j = COS ( B K , BC) AK.AB AK.AC AK.AB AK.AC AK.AC AB AC <=> AK.AB BK.BA BK.BC BK.BA BK.BC iBK.AB ' BK.BC I AB BC Ma AK = ( x - l ; y - 3 ) , B K = (x + 2;y),AB = (-3;-3) nen (*) tuong duong voi < -3(x-l)-3(y-3) -8^^-^)-f^y-'^ 15N/2 37^ 3(x.2).3y 3V2 8^-"'^"^ 2x - y = -1 x - 2 y = -2 x=0 [y = l I5V2 8^ Vay K(0;1). Goi J(a;b) la tam duong tron bang tiep goc A eiia tam giac ABC. Ta co: Phuvng phlip gidi Todn Hinh hoc (ALAB) = (A1AC theo chiiyen dc- AB AC BJ.BC _ BJ.AB BC 5, Phu Khdnh, AJ.AB _ AJ.AC (B],BC) = (BJ,AB Vay J Nguyen ~ 2a - b = - 1 2a + b = - 4 AB Nguyen Tat Cty TNHH Tltu MTV DWH Khang Vie Vi du 1.1.5. T r o n g mat phang v a i he toa do O x y , cho t a m giac A B C biet 5 a = — 4 b = -32 A ( 5 ; 2 ) . P h u o n g t r i n h d u o n g t r u n g true canh BC, d u o n g t r u n g t u y e n C C Ian l u ^ t la x + y - 6 = 0 va 2 x - y + 3 = 0 . T i m toa do cac d i n h B,C cua tam giac A B C . Xgfi gidi. 3^ 4' '2. Goi d : x + y - 6 = 0, C C : 2 x - y + 3 = 0 . Ta c6: C(c;2c + 3) P h u o n g t r i n h BC : x - y + c + 3 = 0 4. Cdc duang ddc hiet trong tam gidc 4.1. D u a n g t r u n g tuyen cua tam giac: K h i gap d u o n g t r u n g tuyen cua tam Goi M la t r u n g d i e m ciia BC, suy ra M : 3-c giac, ta chu yeu khai thac tinh chat d i qua d i n h va t r u n g d i e m cua canh do'i dien. X = - x +y-6 = 0 4.2. D u o n g cao cua tam giac: Ta khai thac t i n h chat d i qua d i n h va v u o n g x-y+c+3=0 goc v o i canh do'i d i e n . y=- 2 c+ 9 4.3. D u o n g t r u n g true cua tam giac: Ta khai thac t i n h chat d i qua t r u n g d i e m va v u o n g goc v o i canh do. Suy ra B ( 3 - 2 c ; 6 - c ) = > C ' ( 4 - c ; 4 - | ) 4.4. D u o n g phan giac t r o n g : Ta khai thac tinh chat ne'u M thuoc A B , M ' d o i x u n g v o i M qua phan giac t r o n g goc A t h i M ' thuoc A C . M a C ' e C C nen ta c6: 2 ( 4 - c ) - ( 4 - - ) + 3 = 0 < = > - - c + 7 = 0 ^ c = — . 2 2 3 Vidu 7 . i . 4 . T r o n g mat ph^ng v o i he tpa do O x y , hay xac d j n h toa do d i n h C cua tam giac A B C bie't rang h i n h chie'u v u o n g goc cua C tren d u o n g thang Vay B A B la d i e m H ( - l ; - l ) , d u o n g phan giac t r o n g cua goc A c6 p h u o n g t r i n h x - y + 2 = 0 va d u o n g cao ke t u B c6 p h u o n g t r i n h 4x + 3y - 1 = 0 . 37 3' 3 • L a p d u o n g thang d d i qua A va v u o n g goc v o i A • Goi H ' la d i e m d o i x u n g v o i H qua d j . K h i do H ' E A C . x + y + 2=::0 x-y+2=0 • D u n g h i n h chie'u v u o n g goc H cua A len A I(-2;0) Ta CO I la t r u n g d i e m ciia H H ' nen H ' ( - 3 ; l ) . D u o n g thang A C d i qua H ' va v u o n g goc v o i d j nen c6 p h u o n g t r i n h : 3 x - 4 y + 13 = 0 . H=dnA 5.2. D u n g A ' d o i x u n g v o i A qua d u o n g thang A Goi A la d u o n g thang d i qua H va v u o n g goc v o i d j . Lay A ' do'i x u n g v o i A qua H : '^A'-^Xj^ x^ lyA'=2yH-yA 5.3. D u n g d u o n g t r o n ( C ) do'i x u n g v o i (C) (c6 tam I , ban k i n h R) qua d u o n g thSng A • D u n g r d o i x u n g v o i I qua d u o n g thang A x-y+2=0 3 x - 4 y + 13 = 0 ' •A(5;7). V i C H d i qua H va v u o n g v o i A H , suy ra p h u o n g t r i n h cua C H : • D u o n g t r o n ( C ) c6 t a m I ' , ban k i n h R. 5.4. D u n g d u o n g thang d ' d o i x u n g v o i d qua d u o n g thang A . • [ 3 x - 4 y + 13 = 0 'if!', r<(.: C h i i y : Giao d i e m ciia (C) va ( C ) chinh la giao d i e m cua va A . • Lay hai d i e m M , N thuoc d . D u n g M ' , N ' Ian l u o t d o i x u n g v o i M , N qua A 3x + 4y + 7 = 0 12 14 5.1. H i n h chie'u v u o n g goc H cua d i e m A len d u o n g thang A K i hi?u d , : X - y + 2 = 0, d2 : 4x + 3y - 1 = 0 . Nen A C n d j = A : 3 '3 , C 5. Mot sobdi todn dung hinh ca ban. JCffigidi P h u o n g t r i n h cua A : x + y + 2 = 0 . Suy ra A n d j = I : 19.4 3 4 d' = M ' N ' . Phumig phdp gidi Todii Uinh hoc theo chuyen dc - Nguyen Vidu 1.1.6.Trong d i e m A(3;2), Pliii Khdnh, Nguyen Cty TNHH Tat Thti m a t p h a n g O x y cho d u o n g thang d : x - 2 y - 3 = 0 va hai 1) T i m toa do true t a m , t a m d u o n g t r o n ngoai tiep t a m giac A B C 2) Viet p h u o n g t r i n h d u a n g thang d ' sao cho d u o n g thang A : 3x + 4y + 1 = 0 2) Viet p h u o n g t r i n h d u o n g t r o n ngoai tiep t a m giac A B C . la d u o n g p h a n giac ciia goc tao b o i hai d u o n g thang d va d ' . 1) Ta tha'y A va B n a m ve m o t phia so v o i d u o n g thang d. G o i A ' la d i e m d o i Jiuang ddn gidi 1) Goi H ( x ; y ) la true t a m t a m giac A B C , ta c6: x u n g v o i A qua d. K h i do v a i m o i d i e m M thuoc d, ta l u o n c6: M A = M A ' Dodo: M A + MB = A ' M + M B > A ' B . D a n g thuc xay ra k h i va chi k h i M = A ' B n d . V i A ' A 1 d nen A A ' c6 p h u o n g t r i n h : 2x + y - 8 = 0 Goi H = d n A A ' = > H : < ^ AH.BC = 0 BH.AC = 0 '(x - 2 ) ( - 7 ) + (y - 1 ) ( - 4 ) = 0 J7x + 4y - 1 8 = 0 (x - 4)(-5) + (y - 3)(-2) = 0 ^ [Sx + 2y - 26 = 0 19 Vay H 2x+y-8=0 34 X = ^ 34 y = 46 - 46 x-2y-3=0 I A 2 = IC2 23 •A' '23 _6 5' 5 yA' = 2 y H - y A = - 5 28 26 5 5^ •(X - 2)2 + (y - 1 ) 2 = (X - 4)2 + (y - 3)2 ( x - 2 ) 2 + ( y - l ) 2 =(x + 3)2+(y + l)2 , do do p h u o n g Vay I X = - x-2y-3=0 13x + 1 4 y - 4 3 = 0 <=> < 5 , J_ •M 10 4 ' 4 rx = l <=><^ , suy ra d n A = I ( l ; - l ) 3x + 4y + l = 0 [ y - - l V i A la p h a n giac cua goc h g p b a i g i i i a hai d u a n g thang d va d ' nen d va F e d ' . Suy ra F I = 14 (2 U 5'5 '3 _16' .5'"5 2 1385 45^ ^ 25^ — N e n no p h u o n g t r i n h la: x + — + y 8 4y V 4, Bai 1 . 1 . 2 . T r o n g m a t p h a n g toa do O x y cho t a m giac A B C c6 A(3;2) ( T i n h toa do cac d i e m B, C. Jiuang la d i e m do'i x i i n g v a i E qua A , ta c6 , d o d o p h u o n g t r i n h d ' : l l x - 2y - 1 3 = 0 . va p h u o n g t r i n h hai d u a n g t r u n g t u y e n B M : 3x + 4y - 3 = 0 , C N : 3x - l O y - 1 7 = 0 . d ' do'i x u n g n h a u qua A , do do l e d ' . Lay E(3;0) G d , ta tim d u g c F V2770 I x-2y-3 =0 2) Xet he p h u o n g t r i n h y = - 25_45 16 J _ 5 '10 45 [8x + 4y = - 5 2) D u o n g t r o n ngoai tiep t a m giac A B C c6 ban k i n h R = l A = 16 25 x= — fx + y = 5 trinh A ' B :]3x + 1 4 y - 4 3 = 0 Nen M : I A 2 = IB2 Goi I ( x ; y ) la t a m d u a n g t r o n ngoai tiep t a m giac A B C , ta c6: V i H la t r u n g d i e m ciia A A ' nen Suy ra A ' B = Viet B(4;3), C ( - 3 ; - l ) CO A ( 2 ; l ) , 1) T i m d i e m M thuoc d u a n g thang d sao cho M A + M B nho nhat, JCffigidi. Khang CP BAI TAP Bai l - l - l - Trong mat phang Oxy cho tam giac A B C B(-l;4). MTV DWH dan gidi :?; • ; Goi G la t r o n g t a m ciia t a m giac, suy ra toa do ciia G la n g h i e m cua he '3x + 4y - 3 = 0 3x-10y-17 = 0 7 ^ = 3 [y = - l > ; r J..J' . I ' i - Phumig phdpgiiii Toan Hitih hoc theo chuyen de- Nguyen Phi'i Khanh, Nguyen Tat Thu Goi E la t r u n g d i e m ciia BC, suy ra EA = - G A => E(2; . 3a + 4 b - 3 = 0 " 3(4-a)-10(-5-b)-17 = 0 Ta CO p h u o n g t r i n h B C : x + 2y + 5 = 0 . a=5 [-3a + 10b + 45 = 0 Tpa d p d i e m C la n g h i e m '^"^ b = -3' [x + y - 2 = 0 [x = 9 L ^ 2y + 5 = 0 ^ |y = - 7 Bai 1.1.3. T r o n g m a t phang toa d o O x y cho t a m giac A B C c6 A ( - 3 ; 0 ) va D o do, ta CO p h u o n g t r i n h A C :2x + y - l l = 0 . p h u o n g t r i n h hai d u o n g phan giac t r o n g B D : x - y - 1 = 0,CE : x + 2y + 1 7 = 0 . Toa d p d i e m A la nghiem ciia he: T i n h toa d o cac d i e m B, C. Jiu&ng ddn gidi Gpi A^ d o i x i i n g v o i A qua BD, suy ra A j e BC va A ^ ( l ; - 4 ) do'i x u n g v o i A qua CE, suy ra A 2 e BC va Vay A A 2 ( - — ; - — ) . 5 Toa d p B la n g h i ^ m cua he: Toa d o C la n g h i e m cua he: Bai x-y-l=0 x = -15 duong cao 3x-4y-19 =0 [ y = -16 x + 2y + 17 = 0 fx = - 3 3x-4y-19 =0 [y = - 7 A A ' : x - y + 2 = 0, 2 2 Jiu&ng •C(-3;-7). duong trung tuyen ddn gidi A(l;2) V i B do'i x i i n g v o i A qua M nen suy ra B = (3; - 2 ) . D u o n g thSng BC d i qua B va v u o n g goc v o i d u o n g thSng: 6x - y - 4 = 0 nen suy ra P h u o n g t r i n h B C : x + 6y + 9 = 0 . Ta CO p h u o n g t r i n h BC: x + y - 2 = 0 x+y-2=0 fx = - l 2x + 5 y - 1 3 = 0 ly = 3 Gpi A ( a ; a + 2 ) , suy ra toa do ciia t r u n g d i e m A C la M va 6x - y - 4 = 0 . Viet p h u o n g trinh |'7x-2y-3 = 0 Toa d o A thoa m a n he: <^ • • [6x-y-4 =0 ddn gidi Suy ra toa d o ciia B la n g h i e m cua he: y =6 d u o n g thang A C . B M : 2x + 5y - 1 3 = 0 . T i n h toa d o cac d i e m A , B. Jiixang 2 => A 5;6 , C ( 9 ; - 7 ) . l u p t CO p h u o n g t r i n h la 7x - 2y - 3 = 0 B(-15;-16). B(-15;-16),C(-3;-7). trinh 2x + y - l l = 0 la t r u n g d i e m cua canh A B . D u o n g t r u n g t u y e n va d u o n g cao qua d i n h A Ian 1 . 1 . 4 . T r o n g m a t phSng toa do O x y cho t a m giac A B C c6 C ( 5 ; - 3 ) va phuong 5 2x-y+l=0 Bai 1.1.6. T r o n g m a t phang v o i h^ tpa d p O x y , cho t a m giac A B C co M (2; 0) 5 Suy ra p h u o n g t r i n h BC : 3x - 4y - 1 9 = 0 . Vay C(9;-7). Gpi B' la d i e m d o i x u n g v o i B qua CE, suy ra B'(5;l) va B' e A C Vay B ( 5 ; - 3 ) , C ( - l ; - 2 ) . Aj Khang Viet Jiic&ng ddn gidi Gia sir B ( a ; b ) , suy ra C ( 4 - a ; - 5 - b ) . T u do ta c6 h^: 3a + 4b - 3 = 0 Cty TNHH MTV DWH Tpa d p t r u n g d i e m N cua BC thoa m a n he: •B(-l;3). + 5 a-1^ '7x-2y-3 = 0 'x+6y+9=0 •N Suy ra A C = 2 . M N = (-4; - 3). P h u o n g t r i n h d u o n g th^ng A C : 3x - 4y + 5 0. ^ ; Bai 1.1.7. T r o n g m a t phSng O x y cho d u o n g t r o n ( C ) : (x - if + (y - 1 ) ^ = 25 . Ma M e B M nen 2 ^ y ^ + 5 ^ - 1 3 = 0 « a = 3 =^ A ( 3 ; 5 ) . 1) L a p p h u o n g t r i n h tiep tuyen cua (C), biet tiep tuyen d i qua A ( 3 ; - 6 ) Vay A ( 3 ; 5 ) , B ( - 1 ; 3 ) . Bai 1.1.5. T r o n g m a t phang toa d p O x y cho tam giac A B C 2) T u d i e m D ( - 4 ; 5 ) ve de'n (C) hai tiep tuyen D M , D N ( M , N la tiep diem). Viet CO B ( l ; —3) va p h u o n g t r i n h d u o n g thang M N . p h u o n g t r i n h d u o n g cao A D : 2 x - y + 1 = 0 , d u o n g phan giac C E : x + y - 2=::0 . T i n h toa d p cac d i e m A , C. ,, J^lurnigd&ngidi D u o n g t r o n (Qxd-taDL.K2i 1), ban k i n h R = 5 . T H U ViEN Tl.VHBtNH THU.AN] (. ,ob « . M ( + Av isH 1 Cty TNHH MTV DWH Phumig phtip giai Toan Hinh hoc theo chuyen dS"- Nguyen Phu Khdnh, Nguyen Tat Thii 1) Gia six A : ax + by + c = 0 la tiep tuyen ciia (C) +) Neu diem M e A : ax + by + c = 0,a ^ 0 thi M Do B e A nen 3a - 6b + c = 0 => c = 6b - 3a A la tiep tuyen ciia (C) nen d(I, A) = R 2a + b + c Va^+b^ = 5<=> -a + 7b =5 -bm-c Khang Viet - ; m , liic nay toa do ciia M chi con mgt an va ta chi can tim mgt phuong trinh. Vi da 1.2A. Trong mat phang Oxy cho duong tron (C): (x - 1 ) ^ + (y - 1 ) ^ = 4 va duong thang A : x - 3 y - 6 = 0. Tim tga dg diem M nam tren A , sao cho tvr M ve dugc hai tiep tuyen M A , MB (A,B la tiep diem) thoa AABM la tam 4 o l 2 a ^ +7ab-12b^ = 0 « a=-ib 3 T u do, ta CO dugc phuong trinh tiep tuyen la: 3x + 4y +15 = 0 va 4x - 3y - 30 = 0 . 2) Goi T ( X ( , ; y Q ) la tiep diem , ta c6: <=> fTe(C) DI.IT = 0 (Xo-2)2+(y„-l)2=25 (xo-4)(xo-2) + (yo4-5)(yo-l) = 0 Xo+yo-4xo-2yo=20 _ , ^ ^2xo-6yo-23 = 0 Xo+yo-6xo+4yo=-3 Vay phuong trinh M N : 2x - 6y - 23 = 0 . § 2. X A C D I N H T O A D O C U A M Q T D I E M Bai toan co ban ciia phuong phap toa do trong mat phang la bai toan xac dinh toa do ciia mot diem. ChSng han, de lap phuong trinh duong thang can tim mot diem di qua va VTPT, voi phuong trinh duong tron thi ta can xac djnh tarn va ban kinh....Chung ta co the gap bai toan tim toa do ciia diem dugc hoi true tiep hoac gian tiep. giac vuong. Xgigiai Duong tron (C) co tam 1(1; 1), ban kinh R = 2 . Vi AAMB vuong va I M la duong phan giac ciia goc AM B nen A M I = 45° Trong tam giac vuong l A M , ta co: I M = 2V2, suy ra M thugc duong tron tam I ban kinh R' = 2 . Mat khac M e A nen M la giao diem ciia A va ( I , R ' ) . Suy ra tga do ciia M la nghiem ciia he x-3y-6=0 x=3y+6 ( x - i ) 2 + ( y - i ) 2 =8 " 'x = 3y + 6 5y^ +14y + 9 = 0 [(3y + 5)2 + ( y - l ) ' =8 y= -l,X= 3 = -^~ =- 5'^ 5 (3 9I Vay CO hai diem M j (3; - l ) va M 2 - ; — thoa yeu cau bai toan. • Ve phuong dien hinh hgc tong hgp thi de xac dinh toa do mot diem, ta Vi du 1.2.2. Trong mat phSng voi he tga do Oxy cho cac duong thang thuong chiing minh diem do thugc hai hinh (H) va (H'). Khi do diem can tim d i : x + y + 3 = 0, d j : x - y - 4 = 0, dg : x - 2 y = 0. Tim tga do diem M nam chinh la giao diem ciia (H) va (H'). tren duong thSng • Ve phuong di^n dai so, de xac dinh toa do ciia mot diem (gom hai toa do) la hai Ian khoang each t u M den duong thang d2 . bai toan di tim hai an. Do do, chiing ta can xac djnh dugc hai phuong trinh chiia hai an va giai he phuong trinh nay ta tim dugc toa do diem can tim. Khi thiet lap phuong trinh chiing ta can luu y: +) Tich v6 huong ciia hai vec to cho ta mgt phuong trinh, +) Hai doan thang bang nhau cho ta mgt phuong trinh, +) Hai vec to bang nhau cho ta hai phuong trinh, 18 ' sao cho khoang each t u M den duong thang d^ bang Xffi gidi -4 3y + 3 Taco M e d 3 , s u y r a M ( 2 y ; y ) . Suy ra d ( M , d i ) = — ^ ; d ( M , d 2 ) = ^ ^ Theo gia thiet ta co: d ( M , d i ) = 2d(M,d2) <^ 3y + 3 ^ 2 l y - 4