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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 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 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 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 ; y i + y ) 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 ; y i - y ) Z=Jx\+y\) u=vc^r^ 6) U V = X ] X + y ] y = > u l v < ; : > u v = 0 \-^\2 + y ] y = • Haivecta u ( x j , y j ) ; v ( x ; y ) 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 ; y ) : 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 : DjCH Vg VAN HOA KHANG V I E T f D i a c h I 71 Dinh Tien Hoang - P Da Kao - Q - 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 CD o AB.CD - • 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 x 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: , Klia Van Can, P H i e p Binh C h a n h , Q Thu Dufc, TP Ho Chi M i n h 'Phuang trinh duong thdng So xuat bSn: 1335 - 201 2/CXB/07 - 21 / D H Q G H N / 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: L K - T N / Q D - N X B D H Q G H N , cap 12/11/2012 In xong va nop luu chieu Q u y I n a m 201 ,, ,^, • n = (a;b) ?t 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 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 ) ^ 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 luon cung phuong voi nhau) axp + byp + c Va^+b^ (phuong trinh duang phdn gidc cua goc tao boi hai duang thdng Cho hai duong thang d^ : a^x + b^y + c^ = va d2 : a j X + b2y + Cj = • Moi quan he giua VTPT va VTCP: n.u = 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 • ~!( + 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: 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 +) MFj = -exp - a va MF2 = -exp + a j, • Dinh: A[(-a;0), A2(a;0), 6^(0;-b) va 62(0; b ) A^A2 = 2a goi la dai true Ion, B]B2 = 2b goi la dai true be ^ Q • Tam sai: e = — = a 2 XQ < M ( x o ; y o ) ( H ) : \ - J ^ = l « ^ - f j = l vataluonco X(,]>a D a • Noi tiep hinh ehir nhat co so PQRS CO ki'ch thuoc 2a va 2b voi b^ = a^ - e^ • • Tieu diem: F|(-c;0), F2(e;0) XQ > a b VI Parabol j. a > ) la mpt Hypebol • Fp F2 : la tieu diem va F|F2 = 2e la tieu eu M ( x ; y ) e ( P ) : MF = x + ^ voi x > B, CAC BAI THlfONG GAP • 1VIF[,MF2 : la eac ban kinh qua tieu § x^ 'Phimng trinh chinh idc cua hypebok a^ y^ = voi h^=c^-a^ Tinh chat vd hlnh dang cua hypebol (fi): cAc BAI T O A N C O B A N 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 phuong • Dinh: Aj(-a;0), A2 (a;0) D Q dai true thuc: 2a va dai true ao: 2b trinh duong thang can lap la: a(x - XQ) + b ( y - yp) = • Tieu diem Fi(-e; 0), Fj ( c; O) • Gia su duong thang can lap A : ax + by + e = Dua vao dieu kien bai toan ta tim dugc a = mb,c = nb Khi phuong trinh A : m x + y + n = Phuong phap • Hai tiem can: y = ± —x a • Hinh eho nhat co so PQRS c6 kieh thuoe 2a, 2b voi b^ = c^ - a^ 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 = Vidu 1.1.1.Trong mat phSng voi he toa 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 ( - ; 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 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 ( ; ) va cac h i n h chie'u ciia A len cac true toa D u o n g tron (C) c6 tam 1(1; ) , ban k i n h R = 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) = 3x + 4y - 36 = 2) Gp i A la tie'p tuye'n can t i m A : a ( x + 6) + b ( y - l ) = 0ax + by + a - b = 0, a^ + b^ 7a+ b G i a s i i ( C ) : x ^ + y ^ - a x - b y + e = •=5o 7a + b = o24a2+14ab-24b2 = o • • a =—b ^ o{7a + b)^ = ( ^ +b^) - a - b + e = -25 - + - - = 0c^ b a=-lb' thay vao n ta c6: — b x + by - 9b = ôã 4x - 3y + 27 = - b + e = -16 [(a - l)(a + 6) + (b - 2)(b - 3) = a^ + b^ - a - b - = a-b a-7b V2 5V2 =- b = -2a,a = 2b thay vao (1) ta CO dugc: (a - if- + 4a^ = - 5a^ - 4a + — = p h u o n g t r i n h v n g h i e m T u ta suy duoc A e A : x - y + 20 = T u o n g t u ta cung c6 dug-c B e A = > A B = A = > A B : x - y + 20 = 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 ; 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 p h u o n g t r i n h Cdch ; 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 = +b a^ + b^ + a - b = = ^ a - b + 20 = d u o n g tron co dang: (x - a ) ^ + ( y - b)^ = 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= 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 - ) 3x + 4y +14 = va 4x - 3y + 27 = =25 a =— •!b = Vay p h u o n g t r i n h (C): x^ + y^ - 3x - 4y = Vay CO hai tie'p tuye'n thoa yeu cau bai toan la: 3) Goi A ( a ; b ) T a c : Ae(C) (a-1)^ + ( b - ) ^ -6a + c = - Do A , A p A e ( C ) nen ta co he: a = ^b thay vao (*) ta c6: - b x + by + - b = o x + y + 14 = lA.NA = va tiep xiic v o i hai A,(3;0), A2(0;4) Va^ + b^ 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 (*) 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 d u o n g thc^ng A, : x - y = va A2 : xXffigidi - y = D o A d i qua N nen p h u o n g trinh c6 dang d(I,A) = • -^''i-' a = 2b thay v a o ( l ) t a c o : ( b - r + b ' ' = - < : : > b = - , a = - o 0 Suy R = D ( I , A , ) = ( Vay p h u o n g t r i n h ( C ) : x I Cac diem, ctqc biet tam Cho t a m giac A B C K h i do: 8l r 4^ ' — + y - , 5j gidc 25 -:l.:J (1) Phumig phdpgidi • Trong tam G ' 7(x-l) + (y-3) = j7x + y-10 = Ma C ' ( - c ; - | ) 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: ( - c ) - ( - - ) + = < = > - - c + = ^ c = — 2 Vidu i T r o n g mat ph^ng v o i he tpa O x y , hay xac d j n h toa 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 + = 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 - = 37 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 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 ' ( - ; 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 : x - y + 13 = 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 x - y + 13 = ' •A(5;7) V i C H d i qua H va v u o n g v o i A H , suy 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 • [ x - y + 13 = 'if!', r H : < ^ AH.BC = BH.AC = '(x - ) ( - ) + (y - ) ( - ) = J7x + 4y - = (x - 4)(-5) + (y - 3)(-2) = ^ [Sx + 2y - 26 = 19 Vay H 2x+y-8=0 34 X = ^ 34 y = 46 - 46 x-2y-3=0 I A = IC2 23 •A' '23 _6 5' yA' = y H - y A = - 28 26 5^ •(X - 2)2 + (y - ) = (X - 4)2 + (y - 3)2 ( x - ) + ( y - l ) =(x + 3)2+(y + l)2 , do p h u o n g Vay I X = - x-2y-3=0 13x + y - = < , J_ •M 10 ' rx = l 0 M e d => M{x; x + 3) o x^ + x - = Trong mat phang Oxy cho diem M(2;3) Viet phuong trinh di qua M(2;3) nen ta suy dupe I nam goc phan t u thii nhat hay Duong tron (C) c6 tarn I ( l ; l ) ban kinh R = CO Bai Gpi I(a; b) la tarn ciia duong tron (C) V i (C) tiep xiic vai hai tryc tpa dp va Jiuang ddn gidi Ta • A ( - l ; 0); B(0; - ) , phuong trinh duong thang A B : x + y + = |a-b a-7b Ta CO (Cj) tiep xiic voi A j , A2 o ^ = (2) Tu (1) va (2) ta c6: 5(a-2)2+5b2=4 a-b = a-7b 5(a-2f+5b2=4 5(a-b) = a - b (I) 65 Phumtg phapgiai Hoac (I) (II) Tonn llhih hoc theo chuyett de- Nguyen ( a - ) +5b^ = b = -2a b ^ - b + 16 = Ban k i n h ( C j ) : R = Y8 Vay K va R = V5 Tat Thu (a;b) = D o ta c6; MTV DWH Khang Viet ^o^+Yo^ - x ( , - x + = (xo+3)(xo-l) + (yo-l)(yo-3) =0 '^o^+yo^-2x0-6x0 + = ''o^+y()^+2xo-4yo=0 5'5 o2xo+yo-3 =0 (1) Tpa d o cac tiep d i e m Tj,T2 thoa m a n dang thuc (1) 2V2 /' Vay p h u o n g t r i n h d u o n g thSng d i qua Tj ,T2 la: 2x + y - = Bai 1.4.11 T r o n g mat phang Oxy cho hai d u o n g thang dj : m x + ( m - l ) y + m = 2^2 va d j : ( m - 2)x - m y + = C h i i n g m i n h rang d i va d2 l u o n cat tai m p t Bai 1.4.9 T r o n g m a t p h a n g v o i he toa d o Decac v u o n g goc O x y , cho hai d u o n g t r o n : ( C i ) : x ^ + y^ - x = v a (C2) :x^ + y^ + x - y - 20 = Viet p h u o n g t r i n h d u o n g tron (C) d i qua cac giao d i e m ciia (Ci), (C2) va c6 tarn n a m tren d u o n g thang A : x + y - = x^ + y ^ - x = diem n a m tren m o t d u o n g t r o n c6' d i n h ; i!M Jihcang dan gidi Ta thay d^ l d nen d j va d2 l u o n cat tai I M a t k h a c : d j d i qua d i e m co'djnh A ( - ; ) , d2 d i qua d i e m c ' d i n h ^ ( ^ ' ^ ) Jiu&ng dan gidi Xet he p h u o n g t r i n h Cty TNHH Ma M T = ( X o + ; y o - l ) ; I T = ( x o - l ; y o - ) v6 n g h i ^ m 'a = 2b a-b Nguyen (11) 5(a - b) = - a + 7b 25a^ - a + 16 = Phil Khdnh, D o d o k h i m thay d o i t h i I l u o n n a m tren d u o n g t r o n d u o n g k i n h A B G i a i he ta d u o c hai cap x^ + y ^ + x - y - = Bai 1.4.12 T r o n g m a t phang v o i he toa d o O x y , cho d u o n g tron ( C ) : x^ + y2 - 2x + 4y = va d u o n g thang d : x - y = T i m toa d p cac d i e m M n g h i e m ( x ; y ) = ( l ; - ) , ( ; ) Suy ( C j ) va (C2) c a t n h a u tai A ( ; - ) , B ( ; ) G o i I la t a m cua (C), ta c6 ( - m ; m ) V i l A = IB nen ta c6: (5 - 6m)^ + ( m + 3)^ = (4 - 6m)^ + ( m - 4)^ o m = - Suy 1(12;-1), R = I A = 5V5 tren d u o n g thSng d, biet tir M ke d u p e hai tiep t u y e n M A , M B den (C) ( A , B la < cac ac tiep d i e m ) va k h o a n g each t u d i e m N ( ; - ) den A B bang —j= v5 Jiuang dan giai Vi M £ d = ^ M ( m ; m ) Vay p h u o n g t r i n h (C): (x -12)2 + (y +1)2 ^ 125 Gpi A ( x o ; y o ) K h i do, ta c6: Bai 1.4.10 T r o n g m a t p h a n g v o i he tpa d p O x y , cho d u o n g t r o n (C) c6 phuong trinh : + y^ - 2x - 6y + = va d i e m M ( - ; l ) G p i T ^ T j la cac tiep d i e m ciia cac tiep t u y e n ke t u M deh ( C ) V i e t p h u o n g t r i n h d u o n g thSng di lA.MA = Ae(C) '^o+yo-2xo+4yo=0 Suy ( m - l)Xo + ( m + quaTpTj Jiu6ng D o do, ta CO p h u o n g t r i n h A B la: ( m - l ) x + ( m + 2)y + m = dan giai D u o n g t r o n (C) c6 t a m 1(1; 3) va ban k i n h R = D o I M = iS 2)yo + m = > R nen diem ^ a t khac: d ( N , A B ) = —j= nen ta c6 p h u o n g t r i n h : M ngoai d u o n g t r o n (C)- G p i T ( X Q ; yg) la tiep d i e m ciia tiep t u y e n ke t u M Ta c6: 66 T€(C) M T IT' T6(C) MT.IT = Giai p h u o n g t r i n h ta t i m d u p e m = 0, m = m-3 ( m - l ) + ( m + 2)2 58 13 Ta loai m = 0, v i k h i d o M ( C ) 67 CtyTNHH Phucntg phdp gidi Tpdti llttih hoc theo chuyen de"-Nguyen Phti Khanh, Nguyen Tat V o i X Q = => yo = Vay C O m o t d i e m M thoa yeu cau bai loan: M 58 58 13 13 Vay qua M ( ; ) va cat ( E ) tai hai d i e m A , B cho M la t r u n g d i e m ciia A B + at / a^ = + bt V ,2 ' -4V3 7' hoac A -4V3 7' ;B 7' Ws' V gal - Cho elip ( E ) : —64 + —9 - \ JIuong / Viet p h u o n g t r i n h tiep t u y e n ( d ) cua va [ y = l + bt2 ^{ y [ b ( t i + t2) = 4(at +1)^ + ( b t +1)^ = 36 ' ^ ( \ 0 \ y + — = o x + 2y - m = 2m m A B tiep xuc v o i ( E ) O) V i t i n h d o i x i i n g nen c6 tiep tuyen thoa m a n la : hay 4x + y - = , ,, x + y - = 0,x + 2y + 10 = , x - y - = , x - y + 10 = + (8a + 18b) t - 23 - Bai x = l + 9t ''ft Phuong trinh A B : doa2+b2>0 4a + 9b = ChQn a = 9,b = -4r=> A : ^ , X c^t,+t2=0 (^4a^ + ddn gidi Do tinh d o i x u n g cua ( E ) nen ta chi can xet t r u o n g h o p tiejp d i e m n a m + at2 Xg = A , B e ( E ) => t j , t la n g h i e m ciia p h u o n g t r i n h X^ y2 T r o n g mat phang O x y , cho elip ( E ) : — + ^ = va d u o n g thSng A : X - V2y + = Gia s u A c3t ( E ) tai B,C T i m d i e m A e A de dien T r o n g mat phang v o i he toa O x y , cho d i e m C ( ; ) v a elip x^ (E): — + ^ 7' J2 S ] ( E ) biet (d) c3t hai true toa O x , O y Ian l u g t tai A , B cho O A = B M la t r u n g d i e m ciia A B k h i va chi k h i a(ti+t2) = ^A + '^B - Bai (2 X ^\ yA=l+bti t j +12 A >0 A , B e Ari> yA + y B = y M Viet A s C loai ddn gidi D u o n g thang A d i qua M , v o i vec to chi p h u o n g u = (a;b) c6 dang: X = i> l 17/ KItang , 48 4V3 V a i xo = - thay vao (1) ta dugc y^ = — X Q = + — y^ Bai Cho elip ( E ) : — + ^ = Viet p h u o n g t r i n h d u o n g thSng d (tj JIu&ng MTV Thu tich tam giac A B C Ion nhat - , J-Iicang ddn gidi = T i m tga cac d i e m A , B thuoc (E) Biet rang A , B do'i xung Gia s u A (x(,; y o ) G ( E ) la d i e m can t i m i\hau qua true h o a n h va t a m giac A B C la tarn giac deu Jiuung ddn gidi Khoang each tir A den ( A ) la A H = , Xo-72y(,+2 -j= G p i A ( X o ; y Q ) , A, B do'i x u n g qua tryc h o a n h nen B ( x Q ; - y o ) BC k h o n g d o i nen S^^^^Q Ion nhat k h i va chi k h i A H I o n nhat :f,' r AB2=4yo2;AC2=(2-x„)'+y(,2 x „ - V y o + < x^,-^/2y„ ViAe(E)^4.>:f = l yo^=i^ (1) A B C la t a m giac deu nen A B - A C o 4yo^ = (XQ - 2)^ + yo^(2) Xo=2 T u (1) va (2) ta co : 7\Q^ - 16xo + = •2 = ^.2V22V2 yo 2V2 + 16 '^o + yo + = =i> A H < 2^3 Phuong phdp gidi Todn Hinh hpc theo chuyen de- Nguyen PM Khdnh, Nguyen Tai Thu 272 Va A H = 2^3 < C t y TNHH MTV DWH / \ cx Xo=2 x„ - ^/2yQ > -i -A(2;-V2) = 3a^ - c ^ = 3.100-4.75 = x = 0=>y = 5;y = - Vay CO hai diem pal l - ' * ' ^ ^ " Vay dien tich tam giac A B C Ian nhat A(2;-\/2 j M( (0; 5); Mj (0; -5) thoa man P^"'°^g ^""'^ chinh tac Hypebol (H), biet (H) di qua va C O tieu diem triing voi tieu diem cua Elip (E):• ^ + ^ = 36 11 Jiuang B a i Trong mat phang vai he toa Oxy, cho elip(F) c6 dai true Ion b5ng 4%/2 , cac dinh tren true nho va cac tieu diem ciing thuoc mot duong Khaug Viet dan gidi Gia su phuong trinh Hypebol (H) c6 dang: =1 t:-u'i ;:,!u.i(i tron Hay lap phuong trinh chinh tac ciia ( E ) Goi ( E ) : — + — = (a > b > O) la phuong trinh elip can tim b Theo gia thiet a = 272, ^ = ^ b - a = a V (1) 9a^ b^ Vi (E) C O tieu diem Fj(5;0) nen cung la tieu diem ciia (H) Vi M e ( H ) : Jiu&ng dan gidi cac dinh tren Oy la B , (0;-b),B2 (0;b), Fi(-c;0), a^ + b^ = 25 a^ = 25 - b^ thay vao (1) ta duoc: 160b2 _ 9(25 - b^) = 9(25 - b^ F2 (c;0) T u giac FjBjF2B2 la hinh thoi nen va c6 dinh nam tren mot duong tron nen F|BjF2B2 la hinh vuong suy b = c ma a^ = b^ + c^ =e> b = phuong trinh: y = x^ - x +1^ = c6 hai tieu diem la F^ va Fj Jiuang M(x; y) e ( E ) , MFi = a + - x; MF2 = a - a a MFj^+MF2^-FjF2^ Ta co: cosFiMF, = i • ^ ^ 2MFj.:.MF, c c + a X -{Icf a + - X J a J o — = o \ / ( ( a + ^- X a - - 70 / y ^ ' va elip(E): y + y^ = Chung minh rang (P) giao (£) tai diem phan biet ciing nam tren mpt duong tron Viet phuong trinh Jiuang ddn gidi dan gidi Hoanh giao diem cua (E) va (P) la nghiem cua phuong trinh Ta c6: a^ = 100,c^ = a^ - b^ = 100 -25 - 75 a « 9b'' - 56b2 - 225 = » b^ = =^ a^ = 16 duong tron di qua diem Tim tpa cac diem M e (E) cho ^ M F j = 120° V x^ ' Bai Trong mat phSng voi h? true toa Oxy cho parabol (P) c6 VayPT ( E ) : — + i - = l ^ ^ B a i Cho Elip ( E ) : ^ Vay phuong trinh cua (H) ' '' — + ( x - x ) - l < = > x ' ' - x ^ + x - - (*) ; ' '' X Xet ham so: f(x) = 9x^ - 36x^ + 37x^ - Taco: f(x) lien tuc tren R c6 f(-l)f(o) B : iyc = y H - y B Do B e BC P( -2; 2) B(b;-4 - b) va P la trung diem BC suy C(-4 - b;b) Mat khac AB C E nen ta co (b - 6)(b + 4) + (b + 10)(b + 3) = o b = 0,b = -6 Taco X G = i ( x A + X B + X c ) , y G = ^ ( y A + y B + y c ) = > A ( ; ) Vay CO hai bp diem thoa yeu cau bai toan: B(0;-4), C(-4;0) hoac B(-6;2), C(2;-6) Vay A ( ; ) , B ( ; - ) , C ( ; ) Vidu i.5.7 Trong mat phSng vai he toa Oxyz, cho tarn giac ABC co dinh A thuoc duong thang d : x - y - = 0, canh BC song song voi d, phuong trinh duong cao B H : x + y + = va trung diem canh AC la M ( l ; l ) VidijL J.5.9 Xac djnh tpa dp dinh B cua tam giac A B C , bie't C(4;3) va cac duong phan giac trong, trung tuyen ke t u A Ian lupt co phucmg trinh x + y - = 0, 4x + 13y-10 = Xffigidi Tim toa cac ctinh A,B,C Gpi C la diem do'i xung cua C qua duong phan giac AD Khi C e AB Xffi gidi Canh AC nam tren duong thang di qua M va vuong goc voi BH Gpi Phuong trinh canh A C : x - y = Mat khac A D co u = (-2; l ) la VTCP va C H u nen ta co: Toa diem A la nghif m cua h^: x-4y-2=0 [x-y =0 x = y = — => A ^ H = ADnCC'=>H(5-2t;t)=>CH = ( l - t ; t - ) CH.U = » -2 ( l - 2t) +1 (t - 3) = o 2' 3' t=1 ' i ' H (3; l ) Do H la trung diem ciia C C , nen C ' ( ; - l ) Vi A = A D n A M ( M la trung diem cua BC) nen tpa dp cua A la nghiem cua Suy toa diem C 3'3 Canh BC di qua C va song song vol duong thang d nen co phuong trinh BC:x-4y+ =0 76 phuong trinh: x + 2y-5 = fx = 4x + y - - [y = -2 A(9;-2) Khi duong thang AB co phuong trinh x + 7y + = nen B (-7t - 5; t) Phucnig phdpgidi Cty TNHH MTV DWH Khang Viet Todii llinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tai Thu Vi M e A M M f -13s+ 10 -;s phuong trinh duong t h i n g A B : x + y + = nen d ( G , A B ) = -13s + 10 = - t - Lai v i M la trung diem ciia BC nen 2s = + t Do d(G,AB) = - - ^ ' C : > •B(-12;l) 4a+ ^ ^ _ _ a =-],a Vi da 1.5.10 Trong mat phang vai h f toa Oxy, cho tarn giac ABC can tai A CO dinh A ( - l ; ) va cac dinh B, C thuoc duong thang A : x - y - = Xac a= =^G dinh toa cac diem B va C, biet dien tich tarn giac ABC bang 18 Xffi gidi Ggi M la trung diein canh BC, tarn giac ABC can tai A nen A M BC Do C ' 2' = - - Xp = ^ , ma • y A + yB + yc = y G yc = "2'2 Suy phuong trinh ciia A M : x + y - = Tuong t u voi a Toa diem M la hghiem ciia h^: x+y-3=0 ,' • ' - ta tim dugc C(-5;0) Vidu J.5.i2.Trong mat phang tpa Oxy, cho tam giac ABC vuong tai A , _7 X =• x-y-4=0 4a+ l M 2' CO dinh C ( - ; l ) , phan giac goc A c6 phuong trinh x + y - = Viet •AM = phuong trinh duong thang BC, biet dien tich tam giac ABC bang 24 va dinh A CO hoanh dp duong Ta c6: S R r = - A M B C = AM.BM = 18 =>BM = — AABC AM = 2^2 Gpi D la diem do'i xiing voi C qua duong thang Matkhac: B G A , s u y r a B ( b ; b - ) nen: d : X + y - = Q, ta tim dugc D(4;9) BM^ =8 b — + b — = h — = 4b = —,b = - 2 2j 2j 2j _5 11 3' ,C B • Voi b = 2' 2'2 V I I I n 11 3^ 11 (3 _5 Voi b = — => B ,C 2' 2'2 • JCffi gidi Vi A thuQC duong tron duong kinh CD nen A la giao diem ciia duong thang d va duong tron duong kinh CD, suy toa dp ciia A la nghi^m ciia he: x+y-5=0 , "^A(4;l) (vixA>0) X + ( y - r =32 2S AABC Vidu 1.5.11 Cho tarn giac A B C voi A ( ; - ) , B ( ; - ) va tarn G thuoc Suy A C = => AB = duong thang d : 3x - y + = Hay tim toa ciia C, biet rang dien tich tarn Vi B thupc duong thang A D : x - = nen B(4; y) giac A B C bang AC = Tu A B - = > ( y - l ) ^ =36=>y = -5,y = JCffigidi Vi AB va A D cung huong nen ta c6 B(4;7) Trung diem I ciia AB la l ( l ; - ) , v i G la tam tam giac ABC nen suy SAGB = | S A B C = => | d (G, AB).AB = =:> d (G, AB) =-1 Vay phuong trinh BC: 3x - 4y +16 = di^ 1.5.13 Cho tam giac ABC c6 M(2;0),N(-l;-l),P(-2;3) Ian lugt la trung diem cac canh AB,BC,CA Tim toa dp cac dinh A , B , C Vi G e d nen suy G(a;3a +1) 78 79 Phucntg i>hapgiai Toihi Hiiih hoc theo chuyen de- Nguyen Phii Khdnh, Nguyen Cty TNHH Tat Thu ^.^^^(a-1)(a.l).b(b.l) Xgigidi ta CO AP = M N hay suy =-5 y^-3 = -\2 T u d o , suy B(l;4),C(3;-4) Vi^t ^.r+(b-2)2 , a ^ - l + (3a + 7)(3a + 8) Thay (1) vao (2) ta co: ^ V(a + l)2+(3a + 8)2 =>A(-5;2) Khang (a-l)a.(b-2)b 7(a + l ) + ( b + l)2 Do M , N , P la trung diem cac canh nen MTV DWH 2a^+9a + l l 2a^+7a + V2a^ +10a + 13 V2a^ +6a + a ^ - a + (3a + 5)(3a+ 7) ^ ^ Va2+(3a + 5)2 HDF = HDE =:> A H la phan giac goc EDF ,BC = 4V2=:>S^ABC = a - b + 4i b^ + 2b = o b = 0,b = -2 a = b - thay vao (1) ta eo: (b - 9)^ + (b -1)^ = 10 v6 nghi^m Vay A(0;4) hoac A ( ; - ) Q Tuang t u , ta c6 BH la phan giac ciia goc DEF Vi du 1.5.19 Trong mat phang tpa dp Oxy, cho tam giac A B C eo dinh Suy H la tam duong tron noi tiep tam giac DEF A ( ; - ) , true tam la H ( ; - l ) , tam duong tron ngoai tiep la I(-2;0) Xac Ta CO: EH.EF EH.ED EF ED FH.FE FH.FD EF FD dinh tpa dp dinh C , biet C c6 hoanh dp duong , giai he ta tim dupe a = l , b = hay H ( l ; l ) Cdch 1:CQ\) la trung diem cua B C , D la diem doi xung voi A qua O Suyra H D = ( l ; - ) nen phuong trinh BC : x - 2y - = HE = ( l ; l ) nen phuong trinh AC : x + y - = HF = (-3; l ) nen phuong trinh AB : 3x - y + = Vi A = A B n A C = ^ A : JCgi gidi 3x - y + = x= -l x+y-4=0 y.5 - •A(-l;5) Tuong tu, ta tim dupe B(-4;-4),C(4;0) Ta CO B H // C D , C H // B D nen t u giac B D C H la hinh binh hanh nen M la trung diem H D Tir suy A H = -2MI: = -2(-2 - x) = -2(-y) s x = -2 y= M(-2;3) Nen duong thang B C qua M c6 A H ( ; ) Vidu i.5.15 Trong mat phang Oxy cho tam giac ABC npi tiep duong tron la vtpt CO phuong trinh la: y - = (C) CO phuong trinh: (x - ) ^ + ( y - l ) ^ =10 Diem M ( ; ) la trung diem Gpi C(a; 3), l A = I C canh BC va dien tich tam giac ABC bang 12 Tim tpa dp cac dinh ciia tam giac ABC o a^ + 4a - = a = - ± V65 => C ( - + ^y65;3) JUffigidL Duong tron (C) c6 tam 1(1; 1), suy M I = (1;-1) V i BC di qua M va vuong goc voi M I nen B C : x - y + = Tpa dp B,C la nghi^m ciia he: 82 (x-l)'+(y-l)'-10 y = x+ x = 2,y = x-y+2=0 x2=4 x = -2,y = 5^ + (-7)^ = (a + 2)^ + (3)^ Cdch Duong tron ngoai tiep tam giac A B C eo phuong trinh (x + 2)^ + y ^ = Phuong trinh A H : x = 3, B C A H B C : y = m ( m ^ -7) Tpa dp B, C la nghiem eiia phuong trinh: (x + 2)2 + m = c ^ x ^ + x + m - = (*) Vi (*) CO hai nghiem, c6 it nhat mpt nghi?m duong nen m A C B H = « m + m - = 0m = Tfr gia thiet ta c6 : AOx = 60°,BOx = 120" =^ AOB=60°; ACB = 30" Vay C(-2 + ^/65;3) Vi diJ 1.5.20 Trong mat phang tpa dp Oxy, cho hai duong CtyTNHH thang dj : Vsx + y = va d j : \/3x - y = Gpi (T) la duong tron tiep xuc voi di tai Nen S,^Bc = ^AB.BC = ^ A B ^ AB^ = ^ ^ AB = v ' Vi A e d j ==> A ( X ; - V X ) , X > ; O A = - ^ A B = - ^ = > A ( - ^ ; - l ) ; A, cat d2 tai hai diem B va C cho tam giac ABC vuong tai B Viet phuong OC = 20A = trinh cua (T), biet tam giac ABC c6 di^n tich bang — va diem A c6 hoanh dp duong Duong tron (T) duong kinh AC c6: I Vi AABC vuong tai B nen AC la duong kinh ciia (T) Gpi ASB = (dj, d j ) = t ta c6 BAC = ASB = t (goc c6 canh tuong ung vuong goc) Gia su ban kinh (T) la R ta c6 : _ BC.BA SAABC" AC sin t AC cost „ „ • , " smtcost V3.N/3 + 1.(-1) Mat khac cos t = ^ l ^ ^ ^ l Phuong trinh (T): x + ^2 / ^;-2 I Vs V3 2.V3' I 2^3 2) -1 Vidu 1.5.21 Trong mat phang Oxy cho tam giac ABC c6 tam G(2;3) Gpi H la true tam cua tam giac ABC Biet duong tron di qua ba trung diem cua ba dean thSng HA,HB, HC c6 phuong trinh : (x -1)^ + (y -1)^ = 10 Viet phuong trinh duong tron ngoai tiep tam giac ABC Xffi gidi Suy S^^g^- = — t u c6 R = Do A e d p C e d j nen A|a;-a\/3 j,c(c;cV3 j them niia vector chi phuong Gpi ( C ) la duong tron (x -1)^ + (y -1)^ = 10, cua d j la U j ( l ; - > ^ ) c6 phuong vuong goc voi A C nen: Mat khac AC = 2R = « 7(c^a)^T(V3(c + a)7 == 2 2^ \ a I 2' Vay phuong trinh cua (T) la 84 sVSa^ ^ x+- , Ciia ba canh, chan ba duong cao va ba trung diem cua cac doan noi true tam voi dinh nam tren mot duong tron c6 tam I , G, Gpi E la tam duong tron ngoai tiep tam Tam duong tron la trung diem ciia AC la: S, ^ "Trong tam ginc, diem gom trung H thang hang va I H = 3IG " o a N / = v i a > nen a = • 'a + c Ta CO ke't qua sau day hinh hoc phang: diem A C u j - 0c-a-3(c + a) = o c = - a I suy ( C ) c6 tam 1(1; 1), ban kinh R = %/lO giac A B C va M la trung diem B C Ta c6: ^/3 , ' 3' { ' 3^ y + - = ^ Phep v j t u V(G,_2): I - > E, M A va M e ( C ) nentaco: E(4;7) va E A = I M = 2N/IO Vay phuong trinh duong tron ngoai tiep tam giac A B C la: ( x - l ) + ( y - ) =40 85 Phuong m phdp giai Toan Hinh hqc theo chuyen de- Nguyen Phii Khdnh, Nguyen Cty TNHH Tat Tliu DWH Khang Viet Jliccmg ddn gidi BAI TAP B a i 1.5.1 Trong mat phSng vai h? tpa dp Oxy, cho tam giac ABC bie't A (5; ) Phuang trinh duang trung true canh BC, duong trung tuyen C C laVi lupt la x + y - = va x - y + = Tim tpa dp cac dinh ciia tam giac ABC Jiimng Gpi d : x - y + = 0; A B : x - y + l = x-y+l=0 Ta c6: A = d n AB => A : • x-2y+3=0 B e AB => B(b;b +1) ddn gidi A(l;2) AB = (b - l ; b - ) Gpi u = (2; 1) la VTCP ciia duong t h i n g d Suy ra: B = (2m - c;9 - 2m - 2c) Vi C la trung diem ciia AB nen: C = ^2m-c + ll-2m-2c^ GCC '2x-y + = x - y + 23 = =>C = 37^ 'U ' yjlih-lf • u.AB _ U.AC Taco: cos^u,ABJ = cos^u,ACJ 2{b-l) + l(b-l) , m - c + l l - m - c + = 0=>m = - - ^ I = ( - - ; — ) Nen 2(-)-2 ^6 6' Phuong trinh BC: 3x - 3y + 23 = Tpa dp cua C la nghif m ciia he: i:' ^ / Do C doi xiing voi B q u a M n e n t a c o C ( - b ; - b ) => AC = (11 - b ; - b ) Gpi C = (c;2c + 3) va I = ( m ; - m ) la trung diem cua BC Tpa dp B '[ MTV AB " AC 2(11-b) + ( - b ) ^J{n-hf +{5-hf 9-b V b - b + 146 b-1 ^l2{h-lf l A(2b + ; - - b ) Ta CO p h u o n g t r i n h A : x + y - = D u o n g thSng M N : 5x + 4y - = , M N = \/41 x+y-2=0 x-y+1=0 Ta CO S ^ A M N = ^ S ^ A B C = I ^ I la t r u n g d i e m cua M N => N ( l ; l ) Ma: d ( A , M N ) = A C d i qua N ( l ; l ) va v u o n g goc v o i d i => A C : 4x - 3y - = A = A C n d2 => A : 4x-3y-l=0 [x = x-y+1=0 y = A(4;5) A B la d u o n g thang d i qua d i e m M ( ; ) nhan M A = (4;3) l a m vec t o chi MN.d(A,MN) = b = -4 b + 31 >(*)A( =7c^ N/4T * 19 (*) 11 13, „ , ; — ) , B(—; ^ 19, ^ , 1 , ), C ( — ; — ) 3 ' ^ •J'iv:' 3^ sir 3' Bai 1.5.7 T r o n g m a t p h i n g v o i he toa Oxy, cho t a m giac A B C can tai A p h u o n g A B : - = ^^—^ 3x - 4y + = B = A B n d j => B : < U'; Jiizang ddn gidi cat A C tai N => N d o i x i i n g v o i M qua d j Goi I = A n d j =i> I : x = -3 3x-4y + = _l=^B(-3; ) 3x + y + 10 = o ' CO d i n h A ( - l ; ) va cac d i n h B, C thuoc d u o n g thang A : x - y - = Xac d j n h toa cac d i e m B va C, biet dien tich tam giac A B C bang 18 Jiu&ng ban k i n h bang V2 nen C thuoc d u o n g t r o n (S) c6 t a m M (S): x^ + (y - 2)^ = A H = d ( A , B C ) - - ^ ; BC = ^ ^ ^ ^ = V , A B = A C = V2 AH Tpa dQ ciia C la n g h i e m cua he p h u o n g t r i n h : 4x-3y-l = y = 4x-l Giai he ta d u o c ( x ; y ) = 4x-l x = l=i>y = l 31 25 33 ^ 25 V i d2 la p h a n giac t r o n g ciia goc A nen B va C nMm khac phia b o la d2 « ( ^ B - YB + l ) ( ' < c - Yc +1) < « 88 (x + l ) + ( y _ ) = AH^ + BC^ '97 97 X-y-4=0 x2+(y-2)2=2 x ^ - x + 31 = Toa d o B,C la n g h i e m ciia h f : v ddn gidi Gpi H la h i n h chieu cua A len A => H la t r u n g d i e m BC V i C each M m o t khoang bang Vi Khang Viet - yc +1 > Vay B 11 2'2 ,C _5 2' ^11 3^ '2 hoac ( x ; y ) = hoac B '3 2' _5^ ,C fll U'2 _5 3' Bai 1.5.8 Cho tam giac A B C n h o n , viet p h u a n g t r i n h d u o n g t h i n g A C , ''iet toa d o chan cac d u o n g cao t u cac d i n h A , B, C Ian l u o t la A j ( - l ; - ) / B:(2;2),q(-l;2) 89 Phuang phiip gidi Todn Hitih hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu Cty TNHH MTVDWH Jiicang dan gidi p a i 1 Cho tam giac ABC c6 phan giac A D : x - y = 0, duong cao (1) CH:2x + y + = 0, AC d i qua M ( ; - ) va AB = A M T i m tpa dp cac dinh Ta CO CBjX = AB,Ci T i i giac A B i H C j npi tiep => ABjCi = A H C j (2) va A H C j = A j H C (3) ciia tam giac ABC Gpi N la diem doi xiing ciia M qua duong phan giac A D N e A B Tir (1), (2), (3) va (4) suy ra: CBjX = AiBjC => AC la phan giac ngoai goc Bj Taco N{-1;0) ciia tarn giac A j B j C j C?inh AB di qua N va vuong goc voi C H => A B : x - 2y +1 = Taco: A^Bj : 4x - 3y - = 0;BiCi : y - = Tpa dp diem A la nghiem ciia h f : Phuong trinh duong phan giac cua goc tao boi hai dt A j B ^ B j C j 2x+y-6=0 Vay phuong trinh canh A C : x + y - = |x=l x-2y + l =0 Tpa dp diem C la nghiem ciia h^ phuong trinh: B a i Trong h^ tpa dp Oxy, cho tarn giac ABC can tai B v d i A ( ; - ) , C(3;5) Diem B nam tren duong thang ( d ) : 2x - y = Viet phuong trinh cac [y = l • Vay A ( l ; l ) JJuang ddn gidi 2x+y+3=0 2x-y-l=0' Taco A M - V ( + 1)2+1 =V5;AB = A M = Taco B e d = > B ( y ; y ) GpiB(2yo-l;yo)eAB Tarn giac ABC can tai B => AB - BC = ^(2y - ) ^ (y - sf « y=| B(|;|) Phuong trinh canh A C : 3x - y - = ' ^ ' AB ^ ^(2yo - 2f + (yo -1)^ = 2Vs « ( y ^ -1)^ = o yo = - , yo = Voi Yo = thi B(5; 3) khong thoa man vi hai diem B, C cimg phia so voi Voi y o = - l thi B ( - ; - l ) thoa man B a i Trong he tpa dp Oxy, cho tam giac ABC c6 dinh A ( ; ) , duong cao ke t u B c6 phuong trinh x - 3y - = va trung tuyen ke tir C c6 phuong trinh x + y +1 = Xac dinh tpa dp cac dinh B,C ciia tam giac Jiucrng ddn gidi Phuong trinh duong cao B H : x - 3y - = va A C : 3x + y - = Vay tpa dp cac dinh ciia tam giac la: A ( l ; l ) , B ( - ; - l ) , C ( - ^ ; - ) Bai Trong mat phang tpa dp Oxy cho diem A ( ; ) , lay d i e m B e O x CO hoanh dp khong am va diem C € Oy c6 tung dp kliong am cho tam giac ABC vuong tai A Tim B, C cho di^n tich tam giac ABC Ion rvhat : i Vi B e B H = > B ( y + ; y ) '3y+_9 y + l ' duong phan giac A D Phuong trinh canh A B : 13x - l l y - 24 = Tpa dp trung diem cua AB la I ^ • y = -2 Vly C( ;-2) duong thang AB va BC « ^(2y -1)^ (y if x-y=0 X V +1 Phuong trinh canh AC : - = ^ ^ - ^ < ; = > x - y - l = x-2y+2=0 y-2 4^+3^ ' Jiudng ddn gidi , v i I thupc trung tuyen ke tit dinhC = = > ^ ^ + ^ + l = o y = -3:^B(-2;-3) 2 ^ ' T u t i m dupe C ( ; - ) 90 _ Jimang ddn gidi Tii giac AjMBjC noi tiep ==> A ^ = A ^ ^ (4) 4x-3y-2 Khang Viet Gpi B ( x ; ) e O x ; C ( ; y ) e O y , x > , y > Ta I CO AB = (x - 2; -1), AC = (-2; y -1) Tam giac ABC vuong tai A suy AB AC = - x - y + = 0r:>y = - x > = > x < 91 Cty TNllll MTV DVVIl Khang Viet Phuang phlip gilii Toan Hinh hpc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu r Dien ti'ch tam giac ABC : S = ^ AB.AC = ^^{x-2f +1.yl4 + {y-lf p^^ B thuoc true hoanh va tam giac A B C vuong nen A(CI;0) AB = ( a - l ; ) , A C = ( ; > / ( a - l ) ) , A B C la tam g i c k va chi A B , A C laiong Cling phuong hay a 5^ T V a i O < x < - = > S = x^-4x + 5a = Vay taco: A(0,1) Phuong trinh duong thang A di qua A c6 dang: ax + by - b = voi a^ + b^ > A la tiep tuyen cua duong tron (I) va chi d(I, A) = R =2 a = 73(a^ + b^) o a^ = 3b^ a = ±73b Suy phuong trinh AB, AC la: ±73x + y - = De tha'y B(1;0) V i Ce A =^ c ( a ; V s ( a - l ) ) 93 ... y 18 -4729^ -1- 37T29 -1 + 37T29 53 + 712 9 y =10 b = 5=>a = 6=>M(6;5) b = -13 Va M 23 + 7T29 _ 51 - 37T29 10 '23 - 712 9 10 ' f - l + 37l29 53 + 7l29 10 10 10 51 + 37l29 -l-37l29 53-7l29 10 10 10 ... lA^ = IB^ Goi I(x;y), taeo: IB^ = I C ^ = 15 16 31 x+y =l X=— 21 y = -: r I x 5^ 8y + o ] Vy o 8j 15 31 16 '16 21 111 ^ — x + —y ^13 = 13 ^ , GI = 13 .13 >GH = -2GI Suy I,G,H thang hang Ta CO4GH... Vg VAN HOA KHANG V I E T f D i a c h I 71 Dinh Tien Hoang - P Da Kao - Q - TP HCN/I ^ Dien thoai: 0873 911 569^^ 3 910 5797 - 3 911 1969 - 3 911 1968 Fax: 08 3 911 0880 ^ XiX2+yiy2 t r o n g d o I la t