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Khi do: 1) u+v = (xi +X2;yi +y2) 2) u-v = (xi-X2;yi-y2) 3) ku=(kxi;kyi) 4) Z=Jx\+y\) u=vc^r^ "''""''^ 6) U.V = X]X2 +y]y2=>ulv<;:>u.v = 0<=> \-^\2 + y]y2 = 0 • Haivecta u(xj,yj); v(x2;y2) ciing phirang vai nhau <=> • Goc giija hai vec to u(xj,yj); v(x2;y2): U.V XiX2+yiy2 cos(u,v)= u V Cho A(x^;y^); B(xB;yB). Khi do : 1) AB = (xB-XA;yB-yA) 2) ^^=^3 = ^{x^ - x + {y ^ - y 3) _XA+XB I ~ trong do I la trung diem ciia AB. • AB 1 CD o AB.CD - 0 • Cho tarn giac ABC voi A{x^;y^), B(xB;yB), C{x^;y^). Khi do trong tarn G(x(,;yg) ciia tarn giac ABC la : V _XA+XB+XC XG- ^ yG= I II. PhirotTg trinh duong thang ,, ,^, 1. 'Phuang trinh duong thdng 1.1. Vec to chi phucmg (VTCP), vec to phdp tuyen (VTPT) cua duong thang: Cho duong thang d. • n = (a;b) ?t 0 goi la vec to phap tuyen cua d neu gia ciia no vuong voi d. 3 Phiam^ phiip giui loiin llinli hoc Iheo chuycn de- Nguyen Phti Khdnh, Nguyen Tat Thti • u =(uj;u2)^0 goi la vec ta chi phuong cua d ne'u gia cua no trung hoac 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) • Moi quan he giua VTPT va VTCP: n.u = 0 . • Ne'u n = (a; b) la mpt VTPT cua duong thang d thi u = (b; -a) la mot VTCP cua duong thang d . • ~ ! ( 5 f • Duong thang AB c6 AB la VTCP. 1.2. Phuwig trinh dumig thang 1.2.1. Phuatig trinh tong qudt cua duong thang: Cho duong thMng d di qua diem A(xQ;yQ) va c6 n = (a;b) la VTPT, khi do phuong trinh tong quat ciia d c6 dang: a(x - XQ) + b(y - yp) = 0. 1.2.2. Phuovg trinh tham so cua duong thang: Cho duong thSng d di qua diem A(xo;yo) va c6 u = (a;b) la VTCP, khi do X = XQ + at phuong trinh tham so cua duong thang d la: , t G R . [y = y(,+bt 2. Vi tri tuang doi giua hai duang thdng. Cho hai duong thcing dj : a^x + bjy + c^ = 0; d2 : a2X + b2y + C2 = 0 . Khi do vi tri |a,x + b,y + Cj =0 tuong doi giua chung phu thuoc vao so nghiem cua h^ : < , (I) [a2X + b2y+ C2 =0 • Neu (I) v6 nghiem thi d^ / /d2 . • Ne'u (I) v6 so nghiem thi d^ = dj • 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 :ajX + 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(XQ;y^). Khi do khoang each tu M den A dugc tinh boi cong thuc: Cty TNHH MTV DWH Khang Viet d(M,(A)): 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 : ajX + b2y + Cj = 0 Phuong trinh phan giac ciia goc tao boi hai duong thang la: - , v , • ajX + b^y + Cj a2X + b2y + C2 + ^/a^^b[ • ,i c-i ; ; . III. Phuang trinh duong tron. rmu.j. 1. <Phuang trinh duang tron: Cho duong tron (C) tam I(a; b), ban kinh R, khi do phuang trinh ciia (C) la: (x-a)2+(y-b)2=R2. Ngoai ra phuong trinh: x^ + y^-2ax-2by + c = 0 voi a^+b^-oO cQng la phuong trinh ciia duong tron c6 tam I(a;b), ban kinh R = Va^ + b^ -c . 2. Phuang trinh tiep tuyen: Cho duong tron (C): (x-a)^+(y-b)^ =R^ • Tiep tuyen A ciia (C) tai diem M la duong thang di qua M va vuong goc vai IM . • Duong thang A : Ax + By + C = 0 la tiep tuyen ciia (C) <=> d(I, A) = R ' • 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 = ^ voi b^=a^-c^. K'. a 2 b^ Vay diem M(xo;y(,) e (E) • = 1 va <a Yo <b, Phumtg phcip giiii Toan Hhih hoc theo chuyen tie- Nguyen Phu Khdnh, Nguyen Tat Thu 3. Tinh chat v>d hlnh dang cua elip: Cho (E): — + ^ = 1, a>b. a b • True doi xung Ox,Oy . Tarn do'i xiing O. 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. • Tieu diem: F|(-c;0), F2(e;0). ^ • Noi tiep trong hinh ehir nhat co so PQRS CO ki'ch thuoc 2a va 2b voi b^ = a^ - e^. Q 0 s Bi R -ex(,. • Tam sai: e = — = < 1 a a a a^ • Hai duong chuan: X = ±—= ± — e e V. Hypebol 1. ^inh nghia: Trong mat phang voi h$ toa do Oxy eho hai diem Fi, F2 eo FjF2 =2c. Tap hop cac diem M ciia mat phSng sao eho MF^ -MFj =2a (2a khong doi va c>a >0) la mpt Hypebol. • Fp F2 : la 2 tieu diem va F|F2 = 2e la tieu eu. • 1VIF[,MF2 : la eac ban kinh qua tieu. 2. 'Phimng trinh chinh idc cua hypebok x^ y^ a^ = 1 voi h^=c^-a^. 3. Tinh chat vd hlnh dang cua hypebol (fi): • True doi xung Ox (true thuc), Oy (true ao). Tam doi xung O . • Dinh: Aj(-a;0), A2 (a;0). DQ dai true thuc: 2a va do dai true ao: 2b. • Tieu diem Fi(-e; 0), Fj ( c; O) . • Hai tiem can: y = ± —x a • Hinh eho nhat co so PQRS c6 kieh thuoe 2a, 2b voi b^ = c^ - a^. • Tam sai: e = — = a • Hai duong chuan: x = ±— = ± — Cty TNHH MTV DWH Khang Viet • DO dai cac ban kinh qua tieu cua M(xo;y(,)e(H): +) MF^ = ex„ + a va MF2 =eX(, - a khi XQ > 0. +) MFj = -exp - a va MF2 = -exp + a khi XQ < 0 . 2 2 .2 2 X(,]>a. . M(xo;yo)6(H):\-J^ = l«^-fj = l vataluonco a D 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. A : duong chuan; F : tieu diem va d(F,A) = p > 0 la tham so'tieu. 2. 'Phuxmg trinh chinh tdc cua ^arabd: = 2px 3.jrinh dang cua Parabol (<P): • True Ox la true do'i xung, dinh O. Tieu diem F(^;0). • Duong chuan A : x = • M(x;y)e(P): MF = x + ^ voi x>0. B, CAC BAI THlfONG GAP § 1. cAc BAI TO AN CO BAN 1. Xg.p phuang trinh duang thang. De lap phuong trinh duong thang A ta thuong dung cac each sau • Tim diemM(xo;yo) ma A di qua va mot VTPT n = (a;b). Khi do phuong trinh duong thang can lap la: a(x - XQ) + b(y- yp) = 0 . • 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:mx + y + n = 0. Phuong phap 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 (C):(x-])2+(y-2)2=25. 1) Viet phuong trinh tiep tuyen ciia (C) tai diem M(4;6), ' 2) Viet phuong trinh tiep tuyen cua (C) xua't phat tu diem N(-6;l) Phucntg phap giai ToAn Ilinh hoc theo chuycn lic- Nguyen Pliii Khanh, Nguyen Tat Thii 3) Tu E(-6;3) ve hai tie'p tuye'n EA, EB (A, B la tie'p diem) den (C). Viet phuong trinh duong thang AB. Duong tron (C) c6 tam 1(1; 2), ban kinh R = 5 . 1) Tie'p tuyen di qua M va vuong goc voi IM nen nhan IM = (3;4) lam VTPT Nen phuong trinh tie'p tuye'n la: 3(x - 4) + 4(y - 6) = 0 <=> 3x + 4y - 36 = 0 . 2) Gpi A la tie'p tuye'n can tim. Do A di qua N nen phuong trinh c6 dang A:a(x + 6) + b(y-l) = 0<=>ax + by + 6a-b = 0, a^ + b^ (*) Ta c6: 7a+ b d(I,A) = Ro Va^ + b^ • = 5o 7a + b = 5^ o{7a + b)^ =25(3^ +b^) o24a2+14ab-24b2 =0o24 - + 12 24 = 0c^ b a = ^b 4 a=-lb' 3 3 3 7 • a=-b thay vao (*) ta c6: -bx + by + -b = 0o3x + 4y+ 14 = 0. 4 4 • a = —b thay vao n ta c6: —bx + by - 9b = 0 «• 4x - 3y + 27 = 0 . 3 3 Vay CO hai tie'p tuye'n thoa yeu cau bai toan la: 3x + 4y +14 = 0 va 4x - 3y + 27 = 0. 3) Goi A(a;b).Tac6: Ae(C) (a-1)^ +(b-2)^ =25 <=> a^ + b^ -2a-4b-20 = 0 a^ + b^ +5a-5b = 0 lA.NA = 0 [(a - l)(a + 6) + (b - 2)(b - 3) = 0 =^7a-b + 20 = 0 Tu do ta suy ra duoc AeA:7x-y + 20 = 0. Tuong tu ta cung c6 dug-c BeA=>AB = A=>AB:7x-y + 20 = 0. 2. Cdch lap phimng trinh dizcrng tron. De lap phuong trinh duong tron (C) ta thuong su dung cac each sau Cdch 7;Tim tam I(a;b) va ban kinh ciia duong tron. Khi do phuong trinh duong tron co dang: (x -a)^ +(y - b)^ = . Cdch 2;Gia su phuong trinh duong tron co dang: x^ + y^ - 2ax - 2by + c = 0 8 Cty TNHH MTV DWH Khang Viet Dua vao gia thie't cua bai toan ta tim dugc a,b,c. Cach nay ta thuong ap dung khi yeu cau viet phuong trinh duong tron di qua ba diem. Vi du 1.1.2. Lap phuong trinh duong tron (C), bie't 1) (C) di qua A(3;4) va cac hinh chie'u ciia A len cac true toa do. 2 2 4 2) (C) CO tam nam tren duong tron (Cj): (x - 2)^ + y = - va tiep xiic voi hai duong thc^ng A, :x-y = 0 va A2 :x-7y = 0. Xffigidi. 1) Goi Ai, A2 Ian iugt la hinh chie'u ciia A len hai true Ox, Oy, suy ra A,(3;0), A2(0;4). Giasii (C):x^+y^-2ax-2by + e = 0. Do A,ApA2e(C) nen ta co he: -6a-8b + e = -25 -6a + c = -9 <=> •! -8b + e = -16 3 a = — 2 b = 2. e = 0 Vay phuong trinh (C): x^ + y^ - 3x - 4y = 0. 774 2) Goi I(a;b) la tam ciia duong tron (C), vi l€(Ci) nen: (a-2) +b =- (1) Do (C) tie'p xuc voi hai duong thing A^Aj nen d(I, Aj) = d(I, A2) a-b a-7b <=>b = -2a,a = 2b V2 5V2 • b = -2a thay vao (1) ta CO dugc: (a - if- + 4a^ = - <=> 5a^ - 4a + — = 0 phuong trinh nay v6 nghiem 9 9 4 4 8 -^''i-' • a = 2b thay vao(l) taco: (2b-2r+b''=-<::>b = -,a = o 00 Suy ra R = D(I,A,) = Vay phuong trinh (C): 3. Cac diem, ctqc biet trong tam gidc. Cho tam giac ABC. Khi do: ( 8l 2 r 4^ ' 8 -:l.:J x — + y I 5j 5 , 25 Phumig phdpgidi Todn Hiith hoc theo chiiyen de - Nguyen Phi't Klidnh, Nguyen Tat Thu • Trong tam G • True tam H : 3 ' 3 AH.BC = 0 BH.AC = 0 Tam duong tron ngoai tiep I: lA^ = IB^ lA^ = IC^ • Tam duong tron noi tiep K : Chu y:C6 the tim K theo each sau: * Gc suy ra D AB.AK AC.AK AB AC BC.BK BA.BK BC AB AB: * Goi D la ehan duong phan giae trong goc A, ta c6: BD = DC , tu day AC AB; * Ta CO AK = KD tu day ta c6 K. BD ^ Tam duong tron bang tiep (goc A) J: AB.AJ AC.AJ AB AC BJ.BC AB.BJ BC AB l?jdui.i.3.Cho tam giac ABC c6 A(1;3),B(-2;0),C 5 3 1) Tim toa do true tam H, tam duong tron ngoai tiep I va trong tam G cua tam giac ABC. Tu dp suy ra I, G, H thang hang; 2) Tim toa do tam duong tron noi tiep va tam duong tron bang tiep goc A cua tam giac ABC. 1) Taeo Yc 1 Xffigidi. 1 9 3 8 Goi H(x;y), suy ra AH = (x-l;y-3),BH = (x + 2;y),BC = 21 3 ,AC = ( 3 _21 8' 8 in CUj TNHH MTV DWH Khang Viet Ma < AH.BC = 0 nen ta eo BH.AC = 0 3 1 7(x-l) + (y-3) = 0 j7x + y-10 = 0 (x + 2) + 7y = 0 [x + 7y + 2 = 0 3 X = — 2 y = -: 1 2' 2 Suy ra H Goi I(x;y), taeo: x + y = l 21 3 111^ — x + —y = 4 4^ 32 lA^ = IB^ IB^ = IC^ (x-l)2+(y-3f =(x + 2)^+y' r 5^ 2 ] = x o + y o I 8y V 8j __15 16 31 15 31 16'16 Ta CO GH = ^13 13^ , GI = 13.13 "l6'l6 >GH = -2GI. Suy ra I,G,H thang hang [) Goi K{x; y) la tam duong tron noi tiep tam giac ABC. Ta c6: KAB = KAC KBC-KBA <=> AK,AB) = (AK,AC = (BK,BC) BK,BA COS(AK, AB) = COS(AK, AC cos (BK, BA j = COS ( BK, BC) <=> < AK.AB AK.AC AK.AB AK.AC AK.AB AK.AC AB AC BK.BA BK.BC BK.BA BK.BC iBK.AB ' BK.BC I AB BC Ma AK = (x-l;y-3),BK = (x + 2;y),AB = (-3;-3) nen (*) tuong duong voi -3(x-l)-3(y-3) -8^^-^)-f^y-'^ 37^ 15N/2 3(x.2).3y 8^-"'^"^ 2x - y = -1 x = 0 x-2y = -2 [y = l 3V2 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 theo chiiyen dc- Nguyen Phu Khdnh, Nguyen Tat Tltu (ALAB) = (A1AC (B],BC) = (BJ,AB AJ.AB _ AJ.AC AC AB BJ.BC _ BJ.AB BC ~ AB 2a - b = -1 2a + b = -4 5 a = — 4 b = -3- 2 Vay J 5, 4' 3^ '2. 4. Cdc duang ddc hiet trong tam gidc 4.1. Duang trung tuyen cua tam giac: Khi gap duong trung tuyen cua tam giac, ta chu yeu khai thac tinh chat di qua dinh va trung diem cua canh do'i dien. 4.2. Duong cao cua tam giac: Ta khai thac tinh chat di qua dinh va vuong goc voi canh do'i dien. 4.3. Duong trung true cua tam giac: Ta khai thac tinh chat di qua trung diem va vuong goc voi canh do. 4.4. Duong phan giac trong: Ta khai thac tinh chat ne'u M thuoc AB, M' doi xung voi M qua phan giac trong goc A thi M' thuoc AC. Vidu 7.i.4.Trong mat ph^ng voi he tpa do Oxy, hay xac djnh toa do dinh C cua tam giac ABC bie't rang hinh chie'u vuong goc cua C tren duong thang AB la diem H(-l;-l), duong phan giac trong cua goc A c6 phuong trinh x-y + 2 = 0 va duong cao ke tuB c6 phuong trinh 4x + 3y -1 = 0 . JCffigidi Ki hi?u d, : X - y + 2 = 0, d2 : 4x + 3y -1 = 0. Goi H' la diem doi xung voi H qua dj. Khi do H' E AC. Goi A la duong thang di qua H va vuong goc voi dj. x + y + 2=::0 Phuong trinh cua A:x + y + 2 = 0. Suy ra A n dj = I: x-y+2=0 I(-2;0) Nen ACndj = A: Ta CO I la trung diem ciia HH' nen H'(-3;l). Duong thang AC di qua H' va vuong goc voi dj nen c6 phuong trinh : 3x-4y + 13 = 0. x-y+2=0 3x-4y + 13 = 0' Vi CH di qua H va vuong voi AH, suy ra phuong trinh cua CH: 3x + 4y + 7=0 [3x-4y + 13 = 0 3 4 •A(5;7). 12 Cty TNHH MTV DWH Khang Vie Vi du 1.1.5. Trong mat phang vai he toa do Oxy , cho tam giac ABC biet A(5; 2). Phuong trinh duong trung true canh BC, duong trung tuyen CC Ian lu^t la x + y- 6 = 0 va 2x-y + 3 = 0. Tim toa do cac dinh B,C cua tam giac ABC. Xgfi gidi. Goi d:x + y-6 = 0, CC: 2x-y + 3 = 0 . Ta c6: C(c;2c + 3) Phuong trinh BC :x-y + c + 3 = 0 Goi M la trung diem ciia BC, suy ra M: 3-c x + y-6 = 0 x-y+c+3=0 X = - y=- 2 c + 9 Suy ra B(3-2c;6-c)=>C'(4-c;4-|) Ma C'eCC nen ta c6: 2(4-c)-(4 ) + 3 = 0<=> c +7 = 0^ c = — . 2 2 3 Vay B 19.4 3 '3 , C 14 37 3' 3 5. Mot sobdi todn dung hinh ca ban. 5.1. Hinh chie'u vuong goc H cua diem A len duong thang A • Lap duong thang d di qua A va vuong goc voi A • H=dnA 5.2. Dung A' doi xung voi A qua duong thang A • Dung hinh chie'u vuong goc H cua A len A Lay A' do'i xung voi A qua H: '^A'-^Xj^ x^ lyA'=2yH-yA 5.3. Dung duong tron (C) do'i xung voi (C) (c6 tam I, ban kinh R) qua duong thSng A • Dung r doi xung voi I qua duong thang A • Duong tron (C) c6 tam I', ban kinh R. Chii y: Giao diem ciia (C) va (C) chinh la giao diem cua va A . 5.4. Dung duong thang d' doi xung voi d qua duong thang A . • Lay hai diem M,N thuoc d. Dung M',N' Ian luot doi xung voi M, N qua A 'if!', r<(.: • d' = M'N'. Phumig phdp gidi Todii Uinh hoc theo chuyen dc - Nguyen Pliii Khdnh, Nguyen Tat Thti Vidu 1.1.6.Trong mat phang Oxy cho duong thang d:x-2y-3 = 0 va hai diem A(3;2), B(-l;4). 1) Tim diem M thuoc duang thang d sao cho MA + MB nho nhat, 2) Viet phuong trinh duang thang d' sao cho duong thang A: 3x + 4y +1 = 0 la duong phan giac ciia goc tao boi hai duong thang d va d'. JCffigidi. 1) Ta tha'y A va B nam ve mot phia so voi duong thang d. Goi A' la diem doi xung voi A qua d. Khi do vai moi diem M thuoc d, ta luon c6: MA = MA' Dodo: MA + MB = A'M + MB>A'B. Dang thuc xay ra khi va chi khi M = A' B n d. Vi A'A 1 d nen AA' c6 phuong trinh: 2x + y -8 = 0 19 2x+y-8=0 Goi H = dnAA'=>H:<^ x-2y-3=0 Vi H la trung diem ciia AA' nen 23 '23 _6 5' 5 •A' yA' = 2yH-yA=-5 Suy ra A'B = 28 26 5 5^ trinh A'B :]3x + 14y-43 = 0 , do do phuong Nen M: x-2y-3=0 13x + 14y-43 = 0 <=> < X = - 16 5 , J_ 10 •M 16 J_ 5 '10 2) Xet he phuong trinh x-2y-3 = 0 rx = l <=><^ , suy ra dnA = I(l;-l) 3x + 4y + l = 0 [y l Vi A la phan giac cua goc hgp bai giiia hai duang thang d va d' nen d va d' do'i xung nhau qua A , do do led'. '3 _16' .5'"5 Lay E(3;0) G d, ta tim dugc F la diem do'i xiing vai E qua A, ta c6 Fed'. Suy ra FI = (2 U 5'5 , do do phuong trinh d': llx - 2y -13 = 0 . 14 Cty TNHH MTV DWH Khang Viet CP BAI TAP Bai l-l-l- Trong mat phang Oxy cho tam giac ABC CO A(2;l), B(4;3), C(-3;-l) 1) Tim toa do true tam, tam duong tron ngoai tiep tam giac ABC 2) Viet phuong trinh duong tron ngoai tiep tam giac ABC. Jiuang ddn gidi 1) Goi H(x;y) la true tam tam giac ABC, ta c6: AH.BC = 0 BH.AC = 0 '(x - 2)(-7) + (y -1)(-4) = 0 J7x + 4y -18 = 0 (x - 4)(-5) + (y - 3)(-2) = 0 ^ [Sx + 2y - 26 = 0 ^ X = 34 y = - 46 Vay H 34 46 Goi I(x;y) la tam duang tron ngoai tiep tam giac ABC, ta c6: •(X - 2)2 + (y -1)2 = (X - 4)2 + (y - 3)2 fx + y = 5 IA2 = IB2 IA2 = IC2 (x-2)2+(y-l)2 =(x + 3)2+(y + l)2 [8x + 4y =-5 x = — 25 y = - 45 Vay I 25_45 4 ' 4 2) Duong tron ngoai tiep tam giac ABC c6 ban kinh R = lA = Nen no phuong trinh la: V2770 ( 25^ 2 45^ ^ 1385 x + — + y- V — I 4, y- V 4y 8 Bai 1.1.2. Trong mat phang toa do Oxy cho tam giac ABC c6 A(3;2) va phuong trinh hai duang trung tuyen BM: 3x + 4y - 3 = 0,CN: 3x - lOy -17 = 0. Tinh toa do cac diem B, C. Jiuang dan gidi : ? ; • ; Goi G la trong tam ciia tam giac, suy ra toa do ciia G la nghiem 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 trung diem ciia BC, suy ra EA = -GA => E(2; . Gia sir B(a;b), suy ra C(4-a;-5-b). Tu do ta c6 h^: 3a + 4b-3 = 0 a = 5 b = -3' 3a + 4b - 3 = 0 " 3(4-a)-10(-5-b)-17 = 0 [-3a + 10b + 45 = 0 Vay B(5;-3),C(-l;-2). Bai 1.1.3. Trong mat phang toa do Oxy cho tam giac ABC c6 A(-3;0) va phuong trinh hai duong phan giac trong BD: x - y -1 = 0,CE : x + 2y +17 = 0 . Tinh toa do cac diem B, C. Jiu&ng ddn gidi Gpi A^ doi xiing voi A qua BD, suy ra Aj e BC va A^(l;-4) Aj do'i xung voi A qua CE, suy ra A2 e BC va A2(-—;-—). 5 5 Suy ra phuong trinh BC : 3x - 4y -19 = 0 . x-y-l=0 Toa dp B la nghi^m cua he: Toa do C la nghiem cua he: x = -15 3x-4y-19 = 0 [y = -16 x + 2y + 17 = 0 fx = -3 B(-15;-16). •C(-3;-7). 3x-4y-19 = 0 [y = -7 Vay B(-15;-16),C(-3;-7). Bai 1.1.4.Trong mat phSng toa do Oxy cho tam giac ABC c6 C(5;-3) va phuong trinh duong cao AA':x-y + 2 = 0, duong trung tuyen BM: 2x + 5y -13 = 0. Tinh toa do cac diem A, B. Jiixang ddn gidi Ta CO phuong trinh BC: x + y - 2 = 0 fx = -l Suy ra toa do ciia B la nghiem cua he: x+y-2=0 2x + 5y-13 = 0 ly = 3 •B(-l;3). Gpi A(a;a + 2), suy ra toa do ciia trung diem AC la M + 5 a-1^ Ma MeBM nen 2^y^ + 5^-13 = 0«a = 3 =^ A(3;5). Vay A(3;5),B(-1;3). Bai 1.1.5. Trong mat phang toa dp Oxy cho tam giac ABC CO B(l; —3) va phuong trinh duong cao AD:2x-y +1 = 0, duong phan giac CE:x + y- 2=::0 .Tinh toa dp cac diem A, C. Cty TNHH MTV DWH Khang Viet Jiic&ng ddn gidi Ta CO phuong trinh BC:x + 2y + 5 = 0. [x + y-2 = 0 [x = 9 Tpa dp diem C la nghiem '^"^ L ^ 2y + 5 = 0 ^ |y = -7 Gpi B' la diem doi xung voi B qua CE, suy ra B'(5;l) va B' e AC Do do, ta CO phuong trinh AC :2x + y- ll = 0. 5 C(9;-7). Toa dp diem A la nghiem ciia he: 2x-y+l=0 2x + y-ll = 0 2 => A y = 6 2 Vay A 5;6 2 ,C(9;-7). A(l;2) Bai 1.1.6. Trong mat phang voi h^ tpa dp Oxy, cho tam giac ABC co M (2; 0) la trung diem cua canh AB. Duong trung tuyen va duong cao qua dinh A Ian lupt CO phuong trinh la 7x - 2y - 3 = 0 va 6x - y - 4 = 0 . Viet phuong trinh duong thang AC. Jiu&ng ddn gidi |'7x-2y-3 = 0 Toa do A thoa man he: <^ • • [6x-y-4 = 0 Vi B do'i xiing voi A qua M nen suy ra B = (3; - 2). Duong thSng BC di qua B va vuong goc voi duong thSng: 6x - y - 4 = 0 nen suy ra Phuong trinh BC: x + 6y + 9 = 0 . '7x-2y-3 = 0 'x+6y+9=0 Tpa dp trung diem N cua BC thoa man he: •N Suy ra AC = 2.MN = (-4; - 3). Phuong trinh duong th^ng AC : 3x - 4y + 5 0. ^ ; Bai 1.1.7. Trong mat phSng Oxy cho duong tron (C): (x - if + (y -1)^ = 25 . 1) Lap phuong trinh tiep tuyen cua (C), biet tiep tuyen di qua A(3;-6) 2) Tu diem D(-4;5) ve de'n (C) hai tiep tuyen DM, DN (M, N la tiep diem). Viet phuong trinh duong thang MN. ,, (. J^lurnigd&ngidi ,ob «.M ( + Duong tron (Qxd-taDL.K2i 1), ban kinh R = 5 . Av isH THU ViEN Tl.VHBtNH THU.AN] 1 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) Do B e A nen 3a - 6b + c = 0 => c = 6b - 3a 2a + b + c A la tiep tuyen ciia (C) nen d(I, A) = R Va^+b^ = 5<=> -a + 7b = 5 ol2a^ +7ab-12b^ = 0« 4 a=-ib 3 Tu do, ta CO dugc phuong trinh tiep tuyen la: 3x + 4y +15 = 0 va 4x - 3y - 30 = 0 . fTe(C) 2) Goi T(X(,;yQ) la tiep diem , ta c6: 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 MN : 2x - 6y - 23 = 0 . § 2. XAC DINH TOA DO CUA MQT DIEM 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. • Ve phuong dien hinh hgc tong hgp thi de xac dinh toa do mot diem, ta thuong chiing minh diem do thugc hai hinh (H) va (H'). Khi do diem can tim chinh la giao diem ciia (H) va (H'). • Ve phuong di^n dai so, de xac dinh toa do ciia mot diem (gom hai toa do) la 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 ' Cty TNHH MTV DWH Khang Viet +) Neu diem M e A : ax + by + c = 0,a ^ 0 thi M -bm-c -;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-3y-6 = 0. Tim tga dg diem M nam tren A , sao cho tvr M ve dugc hai tiep tuyen MA, MB (A,B la tiep diem) thoa AABM la tam giac vuong. Xgigiai Duong tron (C) co tam 1(1; 1), ban kinh R = 2 . Vi AAMB vuong va IM la duong phan giac ciia goc AMB nen AMI = 45° Trong tam giac vuong lAM , ta co: IM = 2V2, suy ra M thugc duong tron tam I ban kinh R' = 2 . Mat khac Me A nen M la giao diem ciia A va (I,R'). Suy ra tga do ciia M la nghiem ciia he x=3y+6 x-3y-6=0 (x-i)2 +(y-i)2 =8 " [(3y + 5)2 +(y-l)' =8 'x = 3y + 6 5y^ +14y + 9 = 0 y = -l,X = 3 = =- ^~ 5'^ 5 (3 9 I Vay CO hai diem Mj (3; -l) va M2 -; — thoa yeu cau bai toan. Vi du 1.2.2. Trong mat phSng voi he tga do Oxy cho cac duong thang di:x + y + 3 = 0, dj :x-y-4 = 0, dg :x-2y = 0. Tim tga do diem M nam tren duong thSng sao cho khoang each tu M den duong thang d^ bang hai Ian khoang each tu M den duong thang d2 . Xffi gidi 3y + 3 -4 Taco Med3,suyra M(2y;y). Suy ra d(M,di) = —^;d(M,d2) = ^^ Theo gia thiet ta co: d(M,di) = 2d(M,d2) <^ 3y + 3 ^2ly-4 [...]... B(4;-1),C(6;3) hoac B(0;3),C(4;5) parabol (P) c6 phuong trinh y^ = x va diem 1(0; 2) Tim toa do hai diem M , N Chti i/.-Ngoai each tren, ta c6 the giai theo each khac nhu sau: thuQC (P) sao cho I M = 4iN Tjnh tien he true toa dp Oxy ve he tuc XAY theo vec to O A , ta c6 cong thuc Jiuong dan gidi Vi M , N e ( P ) nentaco M ( m ^ ; m ) , N ( n ^ n ) S u y r a I M = m ' ^ ; m - 2 , I N = n ^ n - 2 Dodo:... 2 6 5 ' 5 , C ( 0 ; l ) va B 5 ' 5 , C AB = AC tron ( C ) : (x - 1 ) ^ + y^ = 2 va B(2;2) T i m tpa d i e m M thupc d u o n g t r o n (C) sao cho x=ay=l Vay CO hai bp d i e m thoa yeu cau bai toan la: Theo de bai ta c6 he: dat d u p e k h i t = 11=> M ( l ; l ) v5 S Xi.AC = 0 d ( M , AB) Ion nha't t^ - 2 t - 3 -,te(-];3) Suy ra m a x d ( M , A B ) = 5'5 „ 2^5 ^„ AB Suy ra A B = => BC = - — = — ^ 5... i)k>,J 3a-b-4 = - l ( a - l ) 2 + b2 = 2 b = 3a - 3 hoac (a-1)'+(3a-3)2 =2 vi , h ; ^ / , „ b = 3a-3 Vay B ( 3 ; - 1 ) ; C ( 5 ; 3 ) hoac B ( - 1 ; 3 ) , C ( 3 ; 5 ) 21 Phuang phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu 5a^ - 1 6 a + 12 = 0 hoac b = 3a-5 _12 _4 ^i " T ' ' ' " 5 hoac b = 3a-5 5a^ -10a+ 4 = 0 b = 3a-3 I M = 4IN 5 r4 m =4n-2 m - 2 = 4(n - 2) n=l ...Cty TNHH Phuvng plidp gidi Toiin Hiith hoc theo chuyen dc- Nguyen Phii Klidnh, Nguyen MTV DWH Khang Viet Tn't Thu Vi du 1.2.5 Cho parabol (P): y^ = x va hai d i e m A(9; 3), B ( l ; -1) thupc (P) 3y+3=2y-8 3y + 3 = - 2 y + G p i M la d i e... true moi, ta c6 phuong trinh cua dj :X + Y + 2 = 0, d2 :X + Y - 4 = 0 Q : B -> C Vi tam giac ABC vuong can tai A nen phep quay Q (A,±90 ) 22 ; , , , |, '23 Cty TNHH MTV DWH Phumtg phapgiiii Toan Hinh hqc theo chuyen rfe - Nguyen Pht'i Khanh, Nguyen Tat Thu Ma B e Bai 1.2.5 T r o n g mat phSng v o i he true toa do Oxy, cho d u o n g thSng => C e d 1 = Q^^.^^^jo/cli), do do C ^ d2 n d , j x - 3 y - 4 =... M a t khac: a-2b -13 Va M 23 + 7T29 _ 51 - 37T29 10 '23 - 7129 10 ' f - l + 37l29 53 + 7l29 10 10 10 51 + 37l29 -l-37l29 53-7l29 10 10 10 ^75 Cty TNHH MTV DWH Khang Vi?t Phumtg phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phti Khihih, Nguyen Tat Thu Jiuang ddn gidi Q(A-9oO)'*^™P''^""^*""^ ( C i ) : ( x - 2 ) 2 + ( y - 3 ) 2 =13 Toa do diem N la nghiem ciia he: ( x - 2 ) 2 + ( y - 3 ) 2 =13 ( x - ]... 1(2; | ) sao cho dien tich tam giac A B C banglS 26 XQ (X(, XQ Khi do di?n tich tam giac A B C la: S^BC = ^ A B d ( C , A) = 3AB "a = 4 r6-3a^ 2 2 , = 25c* a = 0 Vay hai diem can tim la A(0;1) va B(4;4) Theo gia thiet ta c6: A B = 5 . M(2y;y). Suy ra d(M,di) = —^;d(M,d2) = ^^ Theo gia thiet ta co: d(M,di) = 2d(M,d2) <^ 3y + 3 ^2ly-4 Phuvng plidp gidi Toiin Hiith hoc theo chuyen dc- Nguyen Phii Klidnh, Nguyen. B(0;3),C(4;5). Chti i/ Ngoai each tren, ta c6 the giai theo each khac nhu sau: Tjnh tien he true toa dp Oxy ve he tuc XAY theo vec to OA, ta c6 cong thuc 'x = X + 3 '. Vay diem M(xo;y(,) e (E) • = 1 va <a Yo <b, Phumtg phcip giiii Toan Hhih hoc theo chuyen tie- Nguyen Phu Khdnh, Nguyen Tat Thu 3. Tinh chat v>d hlnh dang cua elip: