1. TUY N T P HNH H C GI I TCH TRONG M T PH NG ( P N CHI TI T) BIN SO N: LU HUY TH NG Ton b ti li u c a th y trang: http://www.Luuhuythuong.blogspot.com H N I, 4/2014 H V TN: L P :. TR NG : 2. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 1 HNH HC GII TCH TRONG MT PHNG Ton b ti liu luyn thi i hc mn ton ca thy Lu Huy Thng: http://www.Luuhuythuong.blogspot.com PHN I NG THNG I. L THUYT CN NH 1. Vect ch phng ca ng thng Vect 0u c gi l vect ch phng ca ng thng nu gi ca n song song hoc trng vi . Nhn xt: Nu u l mt VTCP ca th ku (k 0) cng l mt VTCP ca . Mt ng thng hon ton c xc nh nu bit mt im v mt VTCP. 2. Vect php tuyn ca ng thng Vect 0n c gi l vect php tuyn ca ng thng nu gi ca n vung gc vi . Nhn xt: Nu n l mt VTPT ca th kn (k 0) cng l mt VTPT ca . Mt ng thng hon ton c xc nh nu bit mt im v mt VTPT. Nu u l mt VTCP v n l mt VTPT ca th u n . 3. Phng trnh tham s ca ng thng Cho ng thng i qua 0 0 0( ; )M x y v c VTCP 1 2( ; )u u u= . Phng trnh tham s ca : 0 1 0 2 = + = + x x tu y y tu (1) ( t l tham s). Nhn xt: M(x; y) t R: 0 1 0 2 = + = + x x tu y y tu . Gi k l h s gc ca th: + k = tan, vi = xAv , 0 90 . + k = 2 1 u u , vi 1 0u . 4. Phng trnh chnh tc ca ng thng Cho ng thng i qua 0 0 0( ; )M x y v c VTCP 1 2( ; )u u u= . Phng trnh chnh tc ca : 0 0 1 2 x x y y u u = (2) (u1 0, u2 0). Ch : Trong trng hp u1 = 0 hoc u2 = 0 th ng thng khng c phng trnh chnh tc. 5. Phng trnh tham s ca ng thng PT 0ax by c+ + = vi 2 2 0a b+ c gi l phng trnh tng qut ca ng thng. Nhn xt: Nu c phng trnh 0ax by c+ + = th c: VTPT l ( ; )n a b= v VTCP ( ; )u b a= hoc ( ; )u b a= . Nu i qua 0 0 0( ; )M x y v c VTPT ( ; )n a b= th phng trnh ca l: 0 0( ) ( ) 0a x x b y y + = Cc trng hp c bit: 3. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 2 i qua hai im A(a; 0), B(0; b) (a, b 0): Phng trnh ca : 1 x y a b + = . (phng trnh ng thng theo on chn) . i qua im 0 0 0( ; )M x y v c h s gc k: Phng trnh ca : 0 0( )y y k x x = (phng trnh ng thng theo h s gc) 6. V tr tng i ca hai ng thng Cho hai ng thng 1: 1 1 1 0a x b y c+ + = v 2: 2 2 2 0a x b y c+ + = . To giao im ca 1 v 2 l nghim ca h phng trnh: 1 1 1 2 2 2 0 0 a x b y c a x b y c + + = + + = (1) 1 ct 2 h (1) c mt nghim 1 1 2 2 a b a b (nu 2 2 2, , 0a b c ) 1 // 2 h (1) v nghim 1 1 1 2 2 2 a b c a b c = (nu 2 2 2, , 0a b c ) 1 2 h (1) c v s nghim 1 1 1 2 2 2 a b c a b c = = (nu 2 2 2, , 0a b c ) 7. Gc gia hai ng thng Cho hai ng thng 1: 1 1 1 0a x b y c+ + = (c VTPT 1 1 1( ; )n a b= ) v 2: 2 2 2 0a x b y c+ + = (c VTPT 2 2 2( ; )n a b= ). 0 1 2 1 2 1 2 0 0 1 2 1 2 ( , ) ( , ) 90 ( , ) 180 ( , ) ( , ) 90 n n khi n n n n khi n n = > 1 2 1 2 1 2 1 2 1 2 2 2 2 2 1 2 1 1 2 2 . cos( , ) cos( , ) . . n n a a b b n n n n a b a b + = = = + + Ch : 1 2 1 2 1 2 0a a b b+ = . Cho 1: 1 1y k x m= + , 2: 2 2y k x m= + th: + 1 // 2 k1 = k2 + 1 2 k1. k2 = 1. 8. Khong cch t mt im n mt ng thng Khong cch t mt im n mt ng thng Cho ng thng : 0ax by c+ + = v im 0 0 0( ; )M x y . 0 0 0 2 2 ( , ) ax by c d M a b + + = + V tr tng i ca hai im i vi mt ng thng Cho ng thng : 0ax by c+ + = v hai im ( ; ), ( ; )M M N NM x y N x y . M, N nm cng pha i vi ( )( ) 0M M N Nax by c ax by c+ + + + > . M, N nm khc pha i vi ( )( ) 0M M N Nax by c ax by c+ + + + < . Phng trnh cc ng phn gic ca cc gc to bi hai ng thng Cc h s Phng trnh ng thng Tnh cht ng thng c = 0 i qua gc to O a = 0 // Ox hoc Ox b = 0 // Oy hoc Oy 4. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 3 Cho hai ng thng 1: 1 1 1 0a x b y c+ + = v 2: 2 2 2 0a x b y c+ + = ct nhau. Phng trnh cc ng phn gic ca cc gc to bi hai ng thng 1 v 2 l: 1 1 1 2 2 2 2 2 2 2 1 1 2 2 a x b y c a x b y c a b a b + + + + = + + BI TP C BN HT 1. Cho ng thng : 2 1 0d x y + = . Vit phng trnh ng thng d di dng chnh tc v tham s. Gii Ta c: d c 1 vec-t php tuyn (1; 2)n . Suy ra, d c 1 vec-t ch phng (2;1)u Ta c, d qua ( 1;0)M Vy, phng trnh tham s ca 1 2 : x t d y t = + = Phng trnh chnh tc ca 1 : 2 1 x y d + = HT 2. Cho ng thng 1 : 1 2 x t d y t = + = + . Vit phng trnh ng thng d di dng chnh tc v tng qut. Gii Ta c : d i qua im (1; 1)M v c vec-t ch phng (1;2)u . Suy ra d c 1 vec-t php tuyn (2; 1)n Phng trnh chnh tc ca 1 1 : 1 2 x y d + = Phng trnh tng qut ca : 2( 1) 1.( 1) 0 2 3 0d x y x y + = = HT 3. Cho ng thng 2 1 : 1 2 x y d + = . Vit phng trnh tng qut v tham s ca d . Gii Ta c : d i qua (2; 1)M v nhn vec-t ( 1;2)u lm vec-t ch phng. Suy ra d c 1 vec-t php tuyn (2;1)n Phng trnh tham s ca ng thng 2 : 1 2 x t d y t = = + Phng trnh tng qut ca : 2( 2) 1.( 1) 0 2 3 0d x y x y + + = + = HT 4. Vit phng trnh tng qut ca ng thng d bit : a. Qua (2;1)M nhn (1;2)u lm vec-t ch phng. b. Qua (2;1)M nhn (1;2)n lm vec-t php tuyn. 5. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 4 c. i qua hai im (1;2), ( 2;1)A B d. i qua (1;2)M vi h s gc 2k = Gii a. d c vec-t ch phng (1;2)u suy ra d c 1 vec-t php tuyn (2; 1)n Phng trnh ng thng : 2( 1) 1( 2) 0 2 0d x y x y = = b. Phng trnh ng thng : 1( 2) 2( 1) 0 2 4 0d x y x y + = + = c. Ta c: ( 3; 1)AB = Suy ra ng thng AB c 1 vec-t php tuyn (1; 3)n Vy, phng trnh tng qut ca : 1( 1) 3( 2) 0 3 5 0d x y x y = + = d. Phng trnh ng thng : 2( 1) 2 2 4d y x y x= + = + HT 5. Vit phng trnh ng thng d trong cc trng hp: a. i qua (1;2)M v song song vi ng thng : 2 1 0x y + = b. i qua (1;2)M v vung gc vi ng thng : 2 1 0x y + = Gii a. Ta c: / /d nn phng trnh ng thng : 2 0 ( 1)d x y C C+ + = Mt khc: d qua M nn d c phng trnh: : 2 5 0d x y+ = (tha mn) b. Ta c: d nn d c phng trnh: : 2 0d x y C + = Mt khc, d qua M nn d c phng trnh: : 2 0d x y = BI TP NNG CAO HT 6. Trong mt phng vi h to ,Oxy cho 2 ng thng 1 : 7 17 0d x y + = , 2 : 5 0d x y+ = . Vit phng trnh ng thng d qua im M(0;1) to vi 1 2,d d mt tam gic cn ti giao im ca 1 2,d d . Gii Phng trnh ng phn gic gc to bi d1, d2 l: ( ) ( ) 1 2 2 2 2 2 7 17 5 3 13 0 3 4 0 1 ( 7) 1 1 x y x y x y x y + + + = = = + + ng thng cn tm i qua M(0;1) v song song vi 1 hoc 2 . KL: 3 3 0x y+ = v 3 1 0x y + = http://www.Luuhuythuong.blogspot.com HT 7. Trong mt phng vi h trc to ,Oxy cho cho hai ng thng 1 : 2 5 0d x y + = . 2 : 3 6 7 0d x y+ = . Lp phng trnh ng thng i qua im P(2; 1) sao cho ng thng ct hai ng thng d1 v d2 to ra mt tam gic cn c nh l giao im ca hai ng thng d1, d2. Gii (Cch ny hi c bit v c v rc ri hn so vi HT 6 Bi gii ch mang tnh cht tham kho, nn lm theo 6. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 5 cch HT 6) d1 VTCP 1 (2; 1)a = ; d2 VTCP 2 (3;6)a = Ta c: 1 2. 2.3 1.6 0a a = = nn 1 2d d v d1 ct d2 ti mt im I khc P. Gi d l ng thng i qua P( 2; 1) c phng trnh: : ( 2) ( 1) 0 2 0d A x B y Ax By A B + + = + + = d ct d1, d2 to ra mt tam gic cn c nh I khi d to vi d1 ( hoc d2) mt gc 450 0 2 2 2 2 2 2 2 3 cos 45 3 8 3 0 3 2 ( 1) A B A B A AB B B A A B = = = = + + * Nu A = 3B ta c ng thng : 3 5 0d x y+ = * Nu B = 3A ta c ng thng : 3 5 0d x y = Vy c hai ng thng tho mn yu cu bi ton. : 3 5 0d x y+ = ; : 3 5 0d x y = . HT 8. Trong mt phng ,Oxy cho hai ng thng 1 : 3 5 0d x y+ + = , 2 : 3 1 0d x y+ + = v im (1; 2)I . Vit phng trnh ng thng i qua I v ct 1 2,d d ln lt ti A v B sao cho 2 2AB = . Gii Gi s 1 2( ; 3 5) ; ( ; 3 1)A a a d B b b d ; ( 1; 3 3); ( 1; 3 1)IA a a IB b b= = + I, A, B thng hng 1 ( 1) 3 1 ( 3 3) b k a IB kIA b k a = = + = Nu 1a = th 1b = AB = 4 (khng tho). Nu 1a th 1 3 1 ( 3 3) 3 2 1 b b a a b a + = = 22 2 2 ( ) 3( ) 4 2 2 (3 4) 8AB b a a b t t = + + = + + = (vi t a b= ). 2 2 5 12 4 0 2; 5 t t t t + + = = = + Vi 2 2 0, 2t a b b a= = = = : 1 0x y + + = + Vi 2 2 4 2 , 5 5 5 5 t a b b a = = = = : 7 9 0x y = HT 9. Trong mt phng vi h trc to ,Oxy cho hai ng thng 1 : 1 0d x y+ + = , 2 : 2 1 0d x y = . Lp phng trnh ng thng d i qua M(1;1) ct d1 v d2 tng ng ti A v B sao cho 2 0MA MB+ = . Gii Gi s: A(a; a1), B(b; 2b 1). 7. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 6 T iu kin 2 0MA MB+ = tm c A(1; 2), B(1;1) suy ra : 1 0d x = HT 10. Trong mt phng vi h ta ,Oxy cho im M(1; 0). Lp phng trnh ng thng d i qua M v ct hai ng thng 1 2: 1 0, : 2 2 0d x y d x y+ + = + = ln lt ti A, B sao cho MB = 3MA. Gii 1 2 ( ) ( ; 1 ) ( 1; 1 ) ( ) (2 2; ) (2 3; ) A d A a a MA a a B d B b b MB b b = = . T A, B, M thng hng v 3MB MA= 3MB MA= (1) hoc 3MB MA= (2) (1) 2 1 ; ( ) : 5 1 03 3 ( 4; 1) A d x y B = hoc (2) ( )0; 1 ( ) : 1 0 (4;3) A d x y B = HT 11. Trong mt phng vi h ta ,Oxy cho im M(1; 1). Lp phng trnh ng thng (d) i qua M v ct hai ng thng 1 2: 3 5 0, : 4 0d x y d x y = + = ln lt ti A, B sao cho 2 3 0MA MB = . Gii Gi s 1( ;3 5)A a a d , 2( ;4 )B b b d . V A, B, M thng hng v 2 3MA MB= nn 2 3 (1) 2 3 (2) MA MB MA MB = = + 52( 1) 3( 1) 5 5 (1) ; , (2;2)2 2(3 6) 3(3 ) 2 22 a b a A B a b b = = = = . Suy ra : 0d x y = . + 2( 1) 3( 1) 1 (2) (1; 2), (1;3) 2(3 6) 3(3 ) 1 a b a A B a b b = = = = . Suy ra : 1 0d x = . Vy c : 0d x y = hoc : 1 0d x = . HT 12. Trong mt phng vi h ta ,Oxy Lp phng trnh ng thng d qua (2;1)M v to vi cc trc ta mt tam gic c din tch bng 4S = . Gii Gi ( ;0), (0; ) ( , 0)A a B b a b l giao im ca d vi Ox, Oy, suy ra: : 1 x y d a b + = . Theo gi thit, ta c: 2 1 1 8 a b ab + = = 2 8 b a ab ab + = = . Khi 8ab = th 2 8b a+ = . Nn: 12; 4 : 2 4 0b a d x y= = + = . 8. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 7 Khi 8ab = th 2 8b a+ = . Ta c: 2 4 4 0 2 2 2b b b+ = = . + Vi ( ) ( )2 2 2 : 1 2 2 1 2 4 0b d x y= + + + = + Vi ( ) ( )2 2 2 : 1 2 2 1 2 4 0b d x y= + + + = . Cu hi tng t: a) (8;6), 12M S = . S: : 3 2 12 0d x y = ; : 3 8 24 0d x y + = HT 13. Trong mt phng vi h ta ,Oxy cho im A(2; 1) v ng thng d c phng trnh 2 3 0x y + = . Lp phng trnh ng thng qua A v to vi d mt gc c cos 1 10 = . Gii PT ng thng () c dng: ( 2) ( 1) 0a x b y+ + = 2 0ax by a b+ + = 2 2 ( 0)a b+ Ta c: 2 2 2 1 cos 105( ) a b a b = = + 7a2 8ab + b2 = 0. Chon a = 1 b = 1; b = 7. 1 : 1 0x y + = v 2 : 7 5 0x y + + = http://www.Luuhuythuong.blogspot.com HT 14. Trong mt phng vi h ta ,Oxy cho im (2;1)A v ng thng : 2 3 4 0d x y+ + = . Lp phng trnh ng thng i qua A v to vi ng thng d mt gc 0 45 . Gii PT ng thng () c dng: ( 2) ( 1) 0a x b y+ = (2 ) 0ax by a b+ + = 2 2 ( 0)a b+ . Ta c: 0 2 2 2 3 cos 45 13. a b a b + = + 2 2 5 24 5 0a ab b = 5 5 a b a b = = + Vi 5a b= . Chn 5, 1a b= = Phng trnh : 5 11 0x y + = . + Vi 5a b= . Chn 1, 5a b= = Phng trnh : 5 3 0x y + = . HT 15. Trong mt phng vi h to Oxy , cho ng thng : 2 2 0d x y = v im (1;1)I . Lp phng trnh ng thng cch im I mt khong bng 10 v to vi ng thng d mt gc bng 0 45 . Gii Gi s phng trnh ng thng c dng: ax 0by c+ + = 2 2 ( 0)a b+ . V 0 ( , ) 45d = nn 2 2 2 1 2. 5 a b a b = + 3 3 a b b a = = 9. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 8 Vi 3a b= : 3 0x y c+ + = . Mt khc ( ; ) 10d I = 4 10 10 c+ = 6 14 c c = = Vi 3b a= : 3 0x y c + = . Mt khc ( ; ) 10d I = 2 10 10 c + = 8 12 c c = = Vy cc ng thng cn tm: 3 6 0;x y+ + = 3 14 0x y+ = ; 3 8 0;x y = 3 12 0x y + = . HT 16. Trong mt phng vi h trc ta ,Oxy cho ng thng ( ) : 3 4 0d x y = v ng trn 2 2 ( ) : 4 0C x y y+ = . Tm M thuc (d) v N thuc (C) sao cho chng i xng qua im A(3; 1). Gii M (d) M(3b+4; b) N(2 3b; 2 b) N (C) (2 3b)2 + (2 b)2 4(2 b) = 0 6 0; 5 b b= = Vy c hai cp im: M(4;0) v N(2;2) hoc 38 6 8 4 ; , ; 5 5 5 5 M N HT 17. Trong mat phang toa o ,Oxy cho iem A(1; 1) va ng thng : 2 3 4 0x y+ + = . Tm im B thuc ng thng sao cho ng thng AB v hp vi nhau gc 0 45 . Gii c PTTS: 1 3 2 2 x t y t = = + v VTCP ( 3;2)u = . Gi s (1 3 ; 2 2 )B t t + . 0 ( , ) 45AB = 1 cos( ; ) 2 AB u = . 1 . 2 AB u AB u = 2 15 13169 156 45 0 3 13 t t t t = = = . Vy cc im cn tm l: 1 2 32 4 22 32 ; , ; 13 13 13 13 B B . HT 18. Trong mt phng vi h ta ,Oxy cho ng thng : 3 6 0d x y = v im (3;4)N . Tm ta im M thuc ng thng d sao cho tam gic OMN (O l gc ta ) c din tch bng 15 2 . Gii Ta c (3;4)ON = , ON = 5, PT ng thng ON: 4 3 0x y = . Gi s (3 6; )M m m d+ . Khi ta c 21 ( , ). ( , ) 3 2 ONM ONM S S d M ON ON d M ON ON = = = 4.(3 6) 3 13 3 9 24 15 1; 5 3 m m m m m + = + = = = 10. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 9 + Vi 1 (3; 1)m M= + Vi 13 13 7; 3 3 m M = HT 19. Trong mt phng to ,Oxy cho im (0;2)A v ng thng : 2 2 0d x y + = . Tm trn ng thng d hai im B, C sao cho tam gic ABC vung B v AB = 2BC . Gii Gi s (2 2; ), (2 2; )B b b C c c d . V ABC vung B nn AB d . 0dAB u = 2 6 ; 5 5 B 2 5 5 AB = 5 5 BC = 21 125 300 180 5 BC c c= + = 5 5 1 (0;1) 7 4 7 ; 5 5 5 c C c C = = HT 20. Trong mt phng to ,Oxy cho hai ng thng 1 : 3 0d x y+ = , 2 : 9 0d x y+ = v im (1;4)A . Tm im 1 2,B d C d sao cho tam gic ABC vung cn ti A. Gii Gi 1 2( ;3 ) , ( ;9 )B b b d C c c d ( 1; 1 )AB b b= , ( 1;5 )AC c c= . ABC vung cn ti A . 0AB AC AB AC = = 2 2 2 2 ( 1)( 1) ( 1)(5 ) 0 ( 1) ( 1) ( 1) (5 ) b c b c b b c c + = + + = + (*) V 1c = khng l nghim ca (*) nn (*) 2 2 2 2 2 2 ( 1)(5 ) 1 (1) 1 (5 ) ( 1) ( 1) ( 1) (5 ) (2) ( 1) b c b c c b b c c c + = + + + = + T (2) 2 2 ( 1) ( 1)b c+ = 2b c b c = = . + Vi 2b c= , thay vo (1) ta c 4, 2c b= = (2;1), (4;5)B C . + Vi b c= , thay vo (1) ta c 2, 2c b= = ( 2;5), (2;7)B C . Vy: (2;1), (4;5)B C hoc ( 2;5), (2;7)B C . CC BI TON CC TR HT 21. Trong mt phng vi h to ,Oxy cho im M(3; 1). Vit phng trnh ng thng d i qua M ct cc tia Ox, Oy ti A v B sao cho ( 3 )OA OB+ nh nht. Gii PT ng thng d ct tia Ox ti A(a;0), tia Oy ti B(0;b): 1 x y a b + = (a,b>0) 11. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 10 M(3; 1) d 3 1 3 1 1 2 . 12 C si ab a b a b = + . M 3 3 2 3 12OA OB a b ab+ = + = min 3 6 ( 3 ) 12 3 1 1 2 2 a b a OA OB b a b = = + = == = Phng trnh ng thng d l: 1 3 6 0 6 2 x y x y+ = + = http://www.Luuhuythuong.blogspot.com HT 22. Trong mt phng vi h to ,Oxy vit phng trnh ng thng d i qua im M(1; 2) v ct cc trc Ox, Oy ln lt ti A, B khc O sao cho 2 2 9 4 OA OB + nh nht. Gii ng thng (d) i qua (1;2)M v ct cc trc Ox, Oy ln lt ti A, B khc O, nn ( ;0); (0; )A a B b vi . 0ab Phng trnh ca (d) c dng 1 x y a b + = . V (d) qua M nn 1 2 1 a b + = . p dng bt ng thc Bunhiacpski ta c : 2 2 2 2 1 2 1 3 2 1 9 4 1 . 1. 1 3 9a b a b a b = + = + + + 2 2 9 4 9 10a b + 2 2 9 4 9 10OA OB + . Du bng xy ra khi 1 3 2 : 1 : 3 a b = v 1 2 1 a b + = 20 10, 9 a b= = : 2 9 20 0d x y+ = . HT 23. Trong mt phng vi h ta Oxy , cho im M (0; 2) v hai ng thng 1d , 2d c phng trnh ln lt l 3 2 0x y+ + = v 3 4 0x y + = . Gi A l giao im ca 1d v 2d . Vit phng trnh ng thng i qua M, ct 2 ng thng 1d v 2d ln lt ti B , C (B vC khcA ) sao cho 2 2 1 1 AB AC + t gi tr nh nht. Gii 1 2 ( 1;1)A d d A= . Ta c 1 2d d . Gi l ng thng cn tm. H l hnh chiu vung gc ca A trn . ta c: 2 2 2 2 1 1 1 1 AB AC AH AM + = (khng i) 2 2 1 1 AB AC + t gi tr nh nht bng 2 1 AM khi H M, hay l ng thng i qua M v vung gc vi AM. Phng trnh : 2 0x y+ = . HT 24. Trong mt phng to ,Oxy cho cc im A(0; 1) B(2; 1) v cc ng thng c phng trnh: 12. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 11 1 : ( 1) ( 2) 2 0d m x m y m+ + = ; 2 : (2 ) ( 1) 3 5 0d m x m y m+ + = . Chng minh d1 v d2 lun ct nhau. Gi P = d1 d2. Tm m sao cho PA PB+ ln nht. Gii Xt H PT: ( 1) ( 2) 2 (2 ) ( 1) 3 5 m x m y m m x m y m + = + = + . Ta c 2 1 2 3 1 2 0, 2 1 2 2 m m D m m m m = = + > 1 2,d d lun ct nhau. Ta c: 1 2 1 2(0;1) , (2; 1) ,A d B d d d APB vung ti P P nm trn ng trn ng knh AB. Ta c: 2 2 2 2 ( ) 2( ) 2 16PA PB PA PB AB+ + = = 4PA PB+ . Du "=" xy ra PA = PB P l trung im ca cung AB P(2; 1) hoc P(0; 1) 1m = hoc 2m = . Vy PA PB+ ln nht 1m = hoc 2m = . HT 25. Trong mt phng to ,Oxy cho ng thng (): 2 2 0x y = v hai im ( 1;2)A , (3;4)B . Tm im M () sao cho 2 2 2MA MB+ c gi tr nh nht. Gii Gi s M (2 2; ) (2 3; 2), (2 1; 4)M t t AM t t BM t t+ = + = Ta c: 2 2 2 2 15 4 43 ( )AM BM t t f t+ = + + = 2 min ( ) 15 f t f = 26 2 ; 15 15 M HT 26. Trong mt phng to ,Oxy cho ng thng : 2 3 0d x y + = v 2 im (1;0), (2;1)A B . Tm im M trn d sao cho MA MB+ nh nht. Gii Ta c: (2 3).(2 3) 30 0A A B Bx y x y + + = > A, B nm cng pha i vi d. Gi A l im i xng ca A qua d ( 3;2)A Phng trnh : 5 7 0A B x y + = . Vi mi im M d, ta c: MA MB MA MB A B + = + . M MA MB + nh nht A, M, B thng hng M l giao im ca AB vi d. Khi : 8 17 ; 11 11 M . 13. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 12 PHN II NG TRN Ton b ti liu luyn thi i hc mn ton ca thy Lu Huy Thng: http://www.Luuhuythuong.blogspot.com I. L THUYT CN NH 1. Phng trnh ng trn Phng trnh ng trn c tm I(a; b) v bn knh R: 2 2 2 ( ) ( )x a y b R + = . Nhn xt: Phng trnh 2 2 2 2 0x y ax by c+ + + + = , vi 2 2 0a b c+ > , l phng trnh ng trn tm I(a; b), bn knh R = 2 2 a b c+ . 2. Phng trnh tip tuyn ca ng trn Cho ng trn (C) c tm I, bn knh R v ng thng . tip xc vi (C) ( , )d I R = II. BI TP HT 27. Trong mt phng vi h ta ,Oxy vit phng trnh ng trn tm (2;1)I , bn knh 2R = Gii Phng trnh ng trn: 2 2 ( 2) ( 1) 4x y + = HT 28. Trong mt phng vi h to ,Oxy vit phng trnh ng trn tm (1;2)I v i qua ( 1;1)A Gii Bn knh ng trn: 4 1 5R IA= = + = Phng trnh ng trn cn vit: 2 2 ( 1) ( 2) 5x y + = HT 29. Trong mt phng vi h to ,Oxy vit phng trnh ng trn tm ( 1;3)I v tip xc vi ng thng : 3 4 1 0d x y = Gii Bn knh ng trn: 3 12 1 16 ( , ) 5 5 R d I d = = = Phng trnh ng trn cn vit: 2 2 256 ( 1) ( 3) 25 x y+ + = HT 30. Trong mt phng vi h to ,Oxy vit phng trnh ng trn i qua (1;1), ( 1;3)A B v c bn knh bng 10R = . Gii +) Gi ( ; )I a b l tm ng trn. Ta c, ng trn qua ,A B nn suy ra : 2 2 2 2 (1 ) (1 ) ( 1 ) (3 )IA IB a b a b= + = + 2 2 2 2 1 2 1 2 1 2 9 6a a b b a a b b + + + = + + + + 4 4 8 2a b b a = = + (1) 14. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 13 + Bn knh ng trn : 2 2 10 (1 ) (1 ) 10R IA a b= = + = (2) Thay (1) vo (2) ta c : 2 2 (2) (1 ) ( 1 ) 10a a + = 2 2 1 2 1 2 10a a a a + + + + = 2 2 8a = 2a = +) Vi : 2 4 (2;4)a b I= = Vy, phng trnh ng trn : 2 2 ( 2) ( 4) 10x y + = Vi, 2 0 ( 2;0)a b I= = Vy, phng trnh ng trn : 2 2 ( 2) 10x y+ + = Kt lun : 2 2 ( 2) ( 4) 10x y + = v 2 2 ( 2) 10x y+ + = Vi cu hi tng t : (3;1), (4;0); 13A B R = p s : 2 2 ( 1) ( 2) 13x y + + = v 2 2 ( 6) ( 3) 13x y + = HT 31. Trong mt phng vi h to ,Oxy vit phng trnh ng trn i qua 3 im (3;0), (2;1), ( 1;0)A B C Gii Gi ( ; )I a b l tm ng trn : Ta c : ng trn i qua 3 im A, B, C nn suy ra : 2 2 2 2 IA IB IA IB IC IA IC == = = 2 2 2 2 2 2 2 2 (3 ) (2 ) (1 ) (3 ) ( 1 ) a b a b a b a b + = + + = + 6 9 4 4 2 1 1 6 9 2 1 1 a a b a a a b + = + + = + = + = (1; 1)I Bn knh ng trn : 5R IA= = Vy, phng trnh ng trn : 2 2 ( 1) ( 1) 5x y + + = http://www.Luuhuythuong.blogspot.com HT 32. Trong mt phng vi h to ,Oxy gi A, B l cc giao im ca ng thng (d): 2 5 0x y = v ng trn (C): 2 2 20 50 0x y x+ + = . Hy vit phng trnh ng trn (C) i qua ba im A, B, C(1; 1). Gii Ta giao im ca d v (C) l nghim ca h phng trnh: 2 2 2 2 2 5 0 2 5 20 50 0 (2 5) 20 50 0 x y y x x y x x x x = = + + = + + = 15. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 14 2 3 2 5 2 5 1 3 55 40 75 0 5 5 x y x y x y x xx x x y = = = = = = + = = = Vy, A(3; 1), B(5; 5) ng trn (C) i qua 3 im: (3;1); (5;5); (1;1)A B C Hc sinh lm tng t HT trn ta c: (C): 2 2 4 8 10 0x y x y+ + = HT 33. Trong mt phng vi h to ,Oxy vit phng trnh ng trn i qua hai im (0;4); (1;1)A B v tip xc vi ng thng: : 2 0d x y = Gii Gi ( ; )I a b l tm ng trn Ta c, ng trn i qua 2 im A, B nn suy ra : 2 2 2 2 (0 ) (4 ) (1 ) (1 )IA IB a b a b= + = + 8 16 2 1 2 1b a b + = + + 2 6 14 3 7 (1)a b a b = = ng trn tip xc vi d nn : 2 2 2 ( , ) (4 ) 5 a b IA d I d a b = + = (2) Thay (1) vo (2) ta c : 2 2 7 (3 7) ( 4) 5 b b b + = 2 2 14 49 10 50 65 5 b b b b + + = 2 138 49 236 276 0 49 2 b b b b = + = = Vi, 138 71 49 49 b a= = 71 138 ; 49 49 I ; Bn knh ng trn : 8405 2401 R = Phng trnh ng trn : 2 2 71 138 8405 49 49 2401 x y + = Vi, 2 1 ( 1;2)b a I= = ; Bn knh : 5R = Phng trnh ng trn : 2 2 ( 1) ( 2) 5x y+ + = Kt lun : 2 2 71 138 8405 49 49 2401 x y + = v 2 2 ( 1) ( 2) 5x y+ + = HT 34. Trong mt phng vi h to ,Oxy vit phng trnh ng trn (C) i qua ( 1; 2)A v tip xc vi : 7 5 0d x y = ti im (1;2)M Gii Cch 1 : V ng trn tip xc vi ng thng ti M nn M thuc ng trn. 16. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 15 Nh vy, bi ton tr thnh vit phng trnh ng trn i qua hai im A v M, tip xc vi d. Hc sinh vit tng t HT trn. p s : 2 2 ( 6) ( 3) 50x y+ + = Cch 2 : Gi I l tm ng trn. Ta c, ng trn tip xc vi d ti M nn IM d Phng trnh ng thng : 7 0IM x y c+ + = , IM qua M nn 15c = Vy, : 7 15 0IM x y+ = (15 7 ; )I a a Ta c : ng trn i qua A 2 2 2 2 ( 16 7 ) ( 2 ) ( 14 7 ) (2 )IA IM a a a a = + + = + + 2 2 50 220 260 50 200 200 3a a a a a + = + = Vy, ( 6;3)I , bn knh : 50R = Phng trnh ng trn : 2 2 ( 6) ( 3) 50x y+ + = HT 35. Trong mt phng vi h to ,Oxy vit phng trnh ng trn tip xc vi ng thng : 2 0d x y = ti im (3;1)M v c tm I thuc ng thng 1 : 2 2 0d x y = Gii Ta c: (C) tip xc vi d ti M, suy ra tm I ca (C) thuc ng thng c phng trnh cho bi: (3;1) (1;1) qua M vtpt n : 4 0x y + = Khi : 1 ,I d= ta I l nghim ca h phng trnh: 4 0 (2;2) 2 2 0 x y I x y + = = (C) tip xc vi d khi: 2R MI= = Vy, phng trnh ng trn cn vit: 2 2 ( 2) ( 2) 2x y + = HT 36. Trong mt phng vi h to ,Oxy cho ba ng thng: 1 : 2 3 0d x y+ = , .., 3 : 4 3 2 0d x y+ + = . Vit phng trnh ng trn c tm thuc d1 v tip xc vi d2 v d3. Gii Gi tm ng trn l ( ;3 2 )I t t d1. Khi : 2 3) ( , )( , d I dd I d = 3 4(3 2 ) 5 5 4 3(3 2 ) 2 5 t t t t+ + = + + 2 4 t t = = Vy c 2 ng trn tho mn: 2 2 49 25 ( 2) ( 1)x y = + + v 2 2 9 ( 4) ( 5) 25 x y + + = . Cu hi tng t: a) Vi 1 : 6 10 0d x y = , .., 3 : 4 3 5 0d x y = . 17. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 16 S: 2 2 ( 10) 49x y + = hoc 2 2 2 10 70 7 43 43 43 x y + + = . http://www.Luuhuythuong.blogspot.com HT 37. Trong mt phng vi h to ,Oxy cho hai ng thng : 3 8 0x y+ + = , ' :3 4 10 0x y + = v im A( 2; 1). Vit phng trnh ng trn c tm thuc ng thng , i qua im A v tip xc vi ng thng . Gii Gi s tm ( 3 8; )I t t .. Ta c: ( , )d I IA = 2 2 2 2 3( 3 8) 4 10 ( 3 8 2) ( 1) 3 4 t t t t + = + + + 3t = (1; 3), 5I R = PT ng trn cn tm: 2 2 ( 1) ( 3) 25x y + + = . HT 38. Trong mt phng vi h to ,Oxy cho hai ng thng : 4 3 3 0x y + = v ' : 3 4 31 0x y = . Lp phng trnh ng trn ( )C tip xc vi ng thng ti im c tung bng 9 v tip xc vi '. Tm ta tip im ca ( )C v ' . Gii Gi ( ; )I a b l tm ca ng trn (C). ( )C tip xc vi ti im (6;9)M v ( )C tip xc vi nn 54 34 3 3 3 4 31( , ) ( , ') 4 3 3 6 85 45 5 (3;4) 3( 6) 4( 9) 0 3 4 54 aa b a bd I d I a a IM u a b a b + = + = = = + = + = 25 150 4 6 85 10; 6 54 3 190; 156 4 a a a b a a bb = = = = == Vy: 2 2 ( ) : ( 10) ( 6) 25C x y + = tip xc vi ' ti (13;2)N hoc 2 2 ( ) : ( 190) ( 156) 60025C x y+ + = tip xc vi ' ti ( 43; 40)N HT 39. Trong mt phng vi h to ,Oxy vit phng trnh ng trn i qua (2; 1)A v tip xc vi cc trc to . Gii ng trn tip xc vi cc trc ta nn tm I c dng: 1( ; )I a a hoc 2( ; )I a a Phng trnh ng trn c dng: 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) x a y a a a x a y a a b + + = + = Thay ta im A vo phng trnh ta c: a) 1; 5a a= = b) v nghim. Kt lun: 2 2 ( 1) ( 1) 1x y + + = v 2 2 ( 5) ( 5) 25x y + + = . 18. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 17 HT 40. Trong mt phng vi h ta ,Oxy cho ng thng ( ) : 2 4 0d x y = . Lp phng trnh ng trn tip xc vi cc trc ta v c tm trn ng thng (d). Gii Gi ( ;2 4) ( )I m m d l tm ng trn cn tm. Ta c: 4 2 4 4, 3 m m m m= = = . 4 3 m = th phng trnh ng trn l: 2 2 4 4 16 3 3 9 x y + + = . 4m = th phng trnh ng trn l: 2 2 ( 4) ( 4) 16x y + = . HT 41. Trong mt phng vi h ta ,Oxy cho im A(1;1) v B(3;3), ng thng (): 3 4 8 0x y + = . Lp phng trnh ng trn qua A, B v tip xc vi ng thng (). Gii Tm I ca ng trn nm trn ng trung trc d ca on AB d qua M(1; 2) c VTPT l (4;2)AB = d: 2x + y 4 = 0 Tm I(a;4 2a) Ta c IA = d(I,D) 2 11 8 5 5 10 10a a a = + 2a2 37a + 93 = 0 3 31 2 a a = = Vi a = 3 I(3;2), R = 5 (C): (x 3)2 + (y + 2)2 = 25 Vi a = 31 2 31 ; 27 2 I , R = 65 2 (C): 2 231 4225 ( 27) 2 4 x y + + = HT 42. Trong h to ,Oxy cho hai ng thng : 2 3 0d x y+ = v : 3 5 0x y + = . Lp phng trnh ng trn c bn knh bng 2 10 5 , c tm thuc d v tip xc vi . Gii Tm I d ( 2 3; )I a a + . (C) tip xc vi nn: ( , )d I R = 2 2 10 510 a = 6 2 a a = = (C): 2 2 8 ( 9) ( 6) 5 x y+ + = hoc (C): 2 2 8 ( 7) ( 2) 5 x y + + = . http://www.Luuhuythuong.blogspot.com HT 43. Trong mt phng vi h to ,Oxy cho ng trn (C): 2 2 4 3 4 0x y x+ + = . Tia Oy ct (C) ti A. Lp phng trnh ng trn (C), bn knh R = 2 v tip xc ngoi vi (C) ti A. Gii (C) c tm .., bn knh R= 4; A(0; 2). Gi I l tm ca (C). 19. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 18 PT ng thng IA : 2 3 2 2 x t y t = = + , 'I IA (2 3 ;2 2)I t t + . 1 2 '( 3;3) 2 AI I A t I= = (C): 2 2 ( 3) ( 3) 4x y + = HT 44. Trong mt phng vi h to ,Oxy cho ng trn (C): 2 2 4 5 0x y y+ = . Hy vit phng trnh ng trn (C) i xng vi ng trn (C) qua im M 4 2 ; 5 5 Gii (C) c tm I(0;2), bn knh R = 3. Gi I l im i xng ca I qua M I 8 6 ; 5 5 (C): 2 2 8 6 9 5 5 x y + + = HT 45. Trong mt phng vi h ta ,Oxy cho ng trn (C): 2 2 2 4 2 0x y x y+ + + = . Vit phng trnh ng trn (C) tm M(5; 1) bit (C) ct (C) ti hai im A, B sao cho 3AB = . Gii (C) c tm I(1; 2), bn knh 3R = . PT ng thng IM: x3 4 11 0y = . 3AB = . Gi ( ; )H x y l trung im ca AB. Ta c: 2 2 3 2 H IM IH R AH = = x 2 2 3 4 11 0 9 ( 1) ( 2) 4 y x y = + + = 1 29 ; 5 10 11 11 ; 5 10 x y x y = = = = 1 29 ; 5 10 H hoc 11 11 ; 5 10 H . Vi 1 29 ; 5 10 H . Ta c 2 2 2 43R MH AH = + = PT (C): 2 2 ( 5) ( 1) 43x y + = . Vi 11 11 ; 5 10 H . Ta c 2 2 2 13R MH AH = + = PT (C): 2 2 ( 5) ( 1) 13x y + = . HT 46. Trong mt phng vi h ta ,Oxy cho ng trn (C): 2 2 ( 1) ( 2) 4x y + = v im (3;4)K . Lp phng trnh ng trn (T) c tm K, ct ng trn (C) ti hai im A, B sao cho din tch tam gic IAB ln nht, vi I l tm ca ng trn (C). Gii (C) c tm (1;2)I , bn knh 2R = . IABS ln nht IAB vung ti I 2 2AB = . M 2 2IK = nn c hai ng trn tho YCBT. 20. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 19 + 1( )T c bn knh 1 2R R= = 2 2 1( ) : ( 3) ( 4) 4T x y + = + 2( )T c bn knh 2 2 2 (3 2) ( 2) 2 5R = + = 2 2 1( ) : ( 3) ( 4) 20T x y + = . HT 47. Trong mt phng vi h to ,Oxy vit phng trnh ng trn ni tip tam gic ABC vi cc nh: A(2;3), 1 ;0 , (2;0) 4 B C . Gii im D(d;0) 1 2 4 d < < thuc on BC l chn ng phn gic trong ca gc A khi v ch khi ( ) ( ) 2 2 22 91 3 44 4 1 6 3 1. 2 4 3 d DB AB d d d DC AC d + = = = = + Phng trnh AD: 2 3 1 0 3 3 x y x y + = + = ; AC: 2 3 3 4 6 0 4 3 x y x y + = + = Gi s tm I ca ng trn ni tip c tung l b. Khi honh l 1 b v bn knh cng bng b. V khong cch t I ti AC cng phi bng b nn ta c: ( ) 2 2 3 1 4 6 3 5 3 4 b b b b b + = = + 4 3 5 3 1 3 5 2 b b b b b b = = = = R rng ch c gi tr 1 2 b = l hp l. Vy, phng trnh ca ng trn ni tip ABC l: 2 2 1 1 1 2 2 4 x y + = http://www.Luuhuythuong.blogspot.com HT 48. Trong mt phng to ,Oxy cho hai ng thng (d1): 4 3 12 0x y = v (d2): 4 3 12 0x y+ = . Tm to tm v bn knh ng trn ni tip tam gic c 3 cnh nm trn (d1), (d2) v trc Oy. Gii Gi 1 2 1 2, ,A d d B d Oy C d Oy= = = (3;0), (0; 4), (0;4)A B C ABC cn nh A v AO l phn gic trong ca gc A. Gi I, R l tm v bn knh ng trn ni tip ABC 4 4 ;0 , 3 3 I R = . HT 49. Trong mt phng vi h to ,Oxy cho ng thng d: 1 0x y = v hai ng trn c phng trnh: (C1): 21. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 20 2 2 ( 3) ( 4) 8x y + + = , (C2): 2 2 ( 5) ( 4) 32x y+ + = . Vit phng trnh ng trn (C) c tm I thuc d v tip xc ngoi vi (C1) v (C2). Gii Gi I, I1, I2, R, R1, R2 ln lt l tm v bn knh ca (C), (C1), (C2). Gi s ( ; 1)I a a d . (C) tip xc ngoi vi (C1), (C2) nn 1 1 2 2 1 1 2 2, II R R II R R II R II R= + = + = 2 2 2 2 ( 3) ( 3) 2 2 ( 5) ( 5) 4 2a a a a + + = + + a = 0 I(0; 1), R = 2 Phng trnh (C): 2 2 ( 1) 2x y+ + = . HT 50. Trong mt phng ta ,Oxy cho ng trn ( ) 2 2 : 2 0C x y x+ + = . Vit phng trnh tip tuyn ca ( )C , bit gc gia tip tuyn ny v trc tung bng 30 . Gii 2 2 ( ) : ( 1) 1 ( 1;0); 1C x y I R+ + = = . H s gc ca tip tuyn () cn tm l 3 . PT () c dng 1 : 3 0x y b + = hoc 2 : 3 0x y b + + = + 1 : 3 0x y b + = tip xc (C) 1( , )d I R = 3 1 2 3 2 b b = = + . Kt lun: 1( ) : 3 2 3 0x y + = + 2( ) : 3 0x y b + + = tip xc (C) 2( , )d I R = 3 1 2 3 2 b b = = + . Kt lun: 2( ) : 3 2 3 0x y + + = . HT 51. Trong mt phng vi h to ,Oxy cho ng trn (C): x2 2 6 2 5 0x y y+ + = v ng thng (d): x3 3 0y+ = . Lp phng trnh tip tuyn vi ng trn (C), bit tip tuyn khng i qua gc to v hp vi ng thng (d) mt gc 0 45 . Gii (C) c tm I(3; 1), bn knh R = 5 . Gi s (): ax 0 ( 0)by c c+ + = . T: ( , ) 5 2 cos( , ) 2 d I d = = 2, 1, 10 1, 2, 10 a b c a b c = = = = = = x: 2 10 0 : 2 10 0 y x y = + = . HT 52. Trong h to ,Oxy cho ng trn 2 2 ( ) : ( 1) ( 1) 10C x y + = v ng thng : 2 2 0d x y = . Lp phng trnh cc tip tuyn ca ng trn( )C , bit tip tuyn to vi ng thng d mt gc 0 45 . Gii 22. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 21 (C) c tm (1;1)I bn knh 10R = . Gi ( ; )n a b= l VTPT ca tip tuyn 2 2 ( 0)a b+ , V 0 ( , ) 45d = nn 2 2 2 1 2. 5 a b a b = + 3 3 a b b a = = Vi 3a b= : 3 0x y c+ + = . Mt khc ( ; )d I R = 4 10 10 c+ = 6 14 c c = = Vi 3b a= : 3 0x y c + = . Mt khc ( ; )d I R = 2 10 10 c + = 8 12 c c = = Vy c bn tip tuyn cn tm: 3 6 0;x y+ + = 3 14 0x y+ = ; 3 8 0;x y = 3 12 0x y + = . http://www.Luuhuythuong.blogspot.com HT 53. Trong mt phng vi h to ,Oxy vit phng trnh tip tuyn chung ca hai ng trn (C1): 2 2 2 2 2 0x y x y+ = , (C2): 2 2 8 2 16 0x y x y+ + = . Gii (C1) c tm 1(1; 1)I , bn knh R1 = 2; (C2) c tm 2(4; 1)I , bn knh R2 = 1. Ta c: 1 2 1 23I I R R= = + (C1) v (C2) tip xc ngoi nhau ti A(3; 1) (C1) v (C2) c 3 tip tuyn, trong c 1 tip tuyn chung trong ti A l x = 3 // Oy. * Xt 2 tip tuyn chung ngoi: ( ) : ( ) : 0y ax b ax y b = + + = ta c: 2 2 1 1 2 2 2 2 1 2 2 2 ( ; ) 4 4 ( ; ) 4 1 4 7 2 4 7 2 1 4 4 a b a ad I R a b hay d I R a b b b a b + = = = = + = + + = == + Vy, c 3 tip tuyn chung: 1 2 3 2 4 7 2 2 4 7 2 ( ) : 3, ( ) : , ( ) 4 4 4 4 x y x y x + = = + = + HT 54. Trong mt phng vi h ta ,Oxy cho hai ng trn (C): 2 2 ( 2) ( 3) 2x y + = v (C): 2 2 ( 1) ( 2) 8x y + = . Vit phng trnh tip tuyn chung ca (C) v (C). Gii (C) c tm I(2; 3) v bn knh 2R = ; (C) c tm I(1; 2) v bn knh ' 2 2R = . Ta c: ' 2II R R= = (C) v (C) tip xc trong Ta tip im M(3; 4). V (C) v (C) tip xc trong nn chng c duy nht mt tip tuyn chung l ng thng qua im M(3; 4), c vc t php tuyn l ( 1; 1)II = PTTT: 7 0x y+ = 23. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 22 HT 55. Trong mt phng vi h ta ,Oxy cho hai ng trn 2 2 1( ) : 2 3 0C x y y+ = v x2 2 2( ) : 8 8 28 0C x y y+ + = . Vit phng trnh tip tuyn chung ca 1( )C v 2( )C . Gii 1( )C c tm 1(0;1)I , bn knh 1 2R = ; 2( )C c tm 2(4;4)I , bn knh 2 2R = . Ta c: 1 2 1 25 4I I R R= > = + 1 2( ),( )C C ngoi nhau. Xt hai trng hp: + Nu d // Oy th phng trnh ca d c dng: 0x c+ = . Khi : 1 2 ( , ) ( , ) 4d I d d I d c c= = + 2c = : 2 0d x = . + Nu d khng song song vi Oy th phng trnh ca d c dng: ax:d y b= + . Khi : 1 1 2 ( , ) 2 ( , ) ( , ) d I d d I d d I d = = a 2 2 2 1 2 1 1 4 4 1 1 b a b b a a + = + + + = + + 3 7 ; 4 2 3 3 ; 4 2 7 37 ; 24 12 a b a b a b = = = = = = x: 3 4 14 0d y + = hoc x: 3 4 6 0d y = hoc x: 7 24 74 0d y+ = . Vy: : 2 0d x = ; x: 3 4 14 0d y + = ; x: 3 4 6 0d y = ; x: 7 24 74 0d y+ = . HT 56. Trong mt phng vi h ta ,Oxy cho hai ng trn 2 2 1( ) : 4 5 0C x y y+ = v 2 2 2( ) : 6 8 16 0C x y x y+ + + = . Vit phng trnh tip tuyn chung ca 1( )C v 2( )C . Gii 1( )C c tm 1(0;1)I , bn knh 1 3R = ; 2( )C c tm 2(3; 4)I , bn knh 2 3R = . Gi s tip tuyn chung ca 1 2( ), ( )C C c phng trnh: ax 2 2 0 ( 0)by c a b+ + = + . l tip tuyn chung ca 1 2( ), ( )C C 1 1 2 2 ( , ) ( , ) d I R d I R = = a 2 2 2 2 2 3 (1) 3 4 3 (2) b c a b b c a b + = + + = + T (1) v (2) suy ra 2a b= hoc a3 2 2 b c + = . + TH1: Vi 2a b= . Chn 1b = 2, 2 3 5a c= = x: 2 2 3 5 0y + = + TH2: Vi a3 2 2 b c + = . Thay vo (1) ta c: 2 2 0 2 2 4 3 a a b a b a b = = + = . : 2 0y + = hoc x: 4 3 9 0y = . 24. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 23 HT 57. Trong mt phng ,Oxy cho ng trn (C): 2 2 4 3 4 0x y x+ + = . Tia Oy ct (C) ti im A. Lp phng trnh ng trn (T) c bn knh R = 2 sao cho (T) tip xc ngoi vi (C) ti A. Gii (C) c tm ( 2 3;0)I , bn knh 4R = . Tia Oy ct (C) ti (0;2)A . Gi J l tm ca (T). Phng trnh IA: 2 3 2 2 x t y t = = + . Gi s (2 3 ;2 2) ( )J t t IA+ . (T) tip xc ngoi vi (C) ti A nn A 1 2 ( 3;3) 2 AI J t J= = . Vy: 2 2 ( ) : ( 3) ( 3) 4T x y + = . HT 58. Trong mt phng ,Oxy cho ng trn (C): 2 2 1x y+ = v phng trnh: 2 2 2( 1) 4 5 0x y m x my+ + + = (1). Chng minh rng phng trnh (1) l phng trnh ca ng trn vi mi m. Gi cc ng trn tng ng l (Cm). Tm m (Cm) tip xc vi (C). Gii (Cm) c tm ( 1; 2 )I m m+ , bn knh 2 2 ' ( 1) 4 5R m m= + + + , (C) c tm O(0; 0) bn knh R = 1, OI 2 2 ( 1) 4m m= + + , ta c OI < R Vy (C) v (Cm) ch tip xc trong. R R = OI ( v R > R) 3 1; 5 m m= = . HT 59. Trong mt phng ,Oxy cho cc ng trn c phng trnh 2 2 1 1 ( ) : ( 1) 2 C x y + = v 2 2 2( ) : ( 2) ( 2) 4C x y + = . Vit phng trnh ng thng d tip xc vi 1( )C v ct 2( )C ti hai im ,M N sao cho 2 2MN = . Gii 1( )C c tm 1(1;0)I , bn knh 1 1 2 R = ; 2( )C c tm 1(2;2)I , bn knh 2 2R = . Gi H l trung im ca MN 2 2 2 2 2( , ) 2 2 MN d I d I H R = = = Phng trnh ng thng d c dng: ax 2 2 0 ( 0)by c a b+ + = + . Ta c: 1 2 1 ( , ) 2 ( , ) 2 d I d d I d = = a 2 2 2 2 2 2 2 2 a c a b b c a b + = + + + = + . Gii h tm c a, b, c. Vy: : 2 0; : 7 6 0d x y d x y+ = + = ; : 2 0d x y = ; x: 7 2 0d y = 25. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 24 HT 60. Trong mt phng vi h to ,Oxy cho ng trn (C): 2 2 6 5 0x y x+ + = . Tm im M thuc trc tung sao cho qua M k c hai tip tuyn ca (C) m gc gia hai tip tuyn bng 0 60 . Gii (C) c tm I(3;0) v bn knh R = 2. Gi M(0; m) Oy Qua M k hai tip tuyn MA v MB 0 0 60 (1) 120 (2) AMB AMB = = V MI l phn gic ca AMB nn: (1) AMI = 300 0 sin 30 IA MI = MI = 2R 2 9 4 7m m+ = = (2) AMI = 600 0 sin 60 IA MI = MI = 2 3 3 R 2 4 3 9 3 m + = V nghim Vy c hai im M1(0; 7 ) v M2(0; 7 ) http://www.Luuhuythuong.blogspot.com HT 61. Trong mt phng vi h ta ,Oxy cho ng trn (C) v ng thng nh bi: 2 2 ( ) : 4 2 0; : 2 12 0C x y x y x y+ = + = . Tm im M trn sao cho t M v c vi (C) hai tip tuyn lp vi nhau mt gc 600. Gii ng trn (C) c tm I(2;1) v bn knh 5R = . Gi A, B l hai tip im. Nu hai tip tuyn ny lp vi nhau mt gc 600 th IAM l na tam gic u suy ra R=2 52IM = . Nh th im M nm trn ng trn (T) c phng trnh: 2 2 ( 2) ( 1) 20x y + = . Mt khc, im M nm trn ng thng , nn ta ca M nghim ng h phng trnh: 2 2 ( 2) ( 1) 20 (1) 2 12 0 (2) x y x y + = + = Kh x gia (1) v (2) ta c: ( ) ( ) 2 2 2 3 2 10 1 20 5 42 81 0 27 5 y y y y y y = + + = + = = Vy c hai im tha mn bi l: ( )6;3M hoc 6 27 ; 5 5 M HT 62. Trong mt phng vi h ta ,Oxy cho ng trn (C): 2 2 ( 1) ( 2) 9x y + + = v ng thng : 0d x y m+ + = . Tm m trn ng thng d c duy nht mt im A m t k c hai tip tuyn AB, AC ti 26. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 25 ng trn (C) (B, C l hai tip im) sao cho tam gic ABC vung. Gii (C) c tm I(1; 2), R = 3. ABIC l hnh vung cnh bng 3 3 2IA = 1 5 3 2 1 6 72 m m m m = = = = Cu hi tng t: a) 2 2 ( ) : 1, : 0C x y d x y m+ = + = S: 2m = . HT 63. Trong mt phng vi h ta ,Oxy cho ng trn (C): 2 2 ( 1) ( 2) 9x y + + = v ng thng x: 3 4 0d y m + = . Tm m trn d c duy nht mt im P m t c th k c hai tip tuyn PA, PB ti ng trn (C) (A, B l hai tip im) sao cho PAB l tam gic u. (C) c tm (1; 2)I , bn knh 3R = . PAB u A R2 2 6PI I= = = P nm trn ng trn (T) c tm I, bn knh 6r = . Do trn d c duy nht mt im P tho YCBT nn d l tip tuyn ca (T) 1911 ( , ) 6 6 415 mm d I d m =+ = = = . HT 64. Trong mt phng vi h to ,Oxy cho hai ng trn 2 2 ( ) : 18 6 65 0C x y x y+ + = v 2 2 ( ) : 9C x y + = . T im M thuc ng trn (C) k hai tip tuyn vi ng trn (C), gi A, B l cc tip im. Tm ta im M, bit di on AB bng 4,8 . (C) c tm ( )O 0;0 , bn knh R OA 3= = . Gi H AB OM= H l trung im ca AB AH 12 5 = . Suy ra: OH OA AH2 2 9 5 = = v OA OM OH 2 5= = . Gi s ( ; )M x y . Ta c: M OM 2 2 2 2 ( ) 18 6 65 0 5 25 C x y x y x y + + = = + = 4 5 3 0 x x y y = = = = Vy (4;3)M hoc (5;0)M . HT 65. Trong mt phng vi h to ,Oxy cho ng trn (C): 2 2 ( 1) ( 2) 4x y + + = . M l im di ng trn ng thng : 1d y x= + . Chng minh rng t M k c hai tip tuyn 1MT , 2MT ti (C) (T1, T2 l tip im) v tm to im M, bit ng thng 1 2T T i qua im (1; 1)A . (C) c tm (1; 2)I , bn knh 2R = . Gi s 0 0( ; 1)M x x d+ . 2 2 2 0 0 0( 1) ( 3) 2( 1) 8 2IM x x x R= + + = + + > = M nm ngoi (C) qua M k c 2 tip tuyn ti (C). 27. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 26 Gi J l trung im IM 0 01 1 ; 2 2 x x J + . ng trn (T) ng knh IM c tm J bn knh 1 2 IM R = c phng trnh 2 2 2 2 0 0 0 01 1 ( 1) ( 3) ( ) : 2 2 4 x x x x T x y + + + + = T M k c 2 tip tuyn MT1, MT2 n (C) 0 1 2 1 290 , ( )IT M IT M T T T= = 1 2{ , } ( ) ( )T T C T = to 1 2,T T tho mn h: 2 2 2 20 0 0 0 0 0 0 2 2 1 1 ( 1) ( 3) ( ) ( ) (1 ) (3 ) 3 0 (1)2 2 4 ( 1) ( 2) 4 x x x x x y x x x y x x y + + + + = + = + + = To cc im 1 2,T T tho mn (1), m qua 2 im phn bit xc nh duy nht 1 ng thng nn phng trnh 1 2T T l 0 0 0(1 ) (3 ) 3 0x x y x x + = . (1; 1)A nm trn 1 2T T nn 0 0 01 (3 ) 3 0x x x + + = 0 1x = (1;2)M . HT 66. Trong mt phng vi h ta ,Oxy cho ng trn (C): 2 2 ( 1) ( 1) 25x y+ + = v im M(7; 3). Lp phng trnh ng thng (d) i qua M ct (C) ti hai im A, B phn bit sao cho MA = 3MB. /( ) 27 0M CP = > M nm ngoi (C). (C) c tm I(1;1) v R = 5. Mt khc: 2 /( ) . 3 3 3M CP MAMB MB MB BH= = = = 2 2 4 [ ,( )]IH R BH d M d = = = Ta c: pt(d): a(x 7) + b(y 3) = 0 (a2 + b2 > 0). 2 2 06 4 [ ,( )] 4 4 12 5 aa b d M d a ba b = = = = + . Vy (d): y 3 = 0 hoc (d): 12x 5y 69 = 0. HT 67. Trong mt phng vi h to ,Oxy lp phng trnh ng thng d i qua im A(1; 2) v ct ng trn (C) c phng trnh 2 2 ( 2) ( 1) 25x y + + = theo mt dy cung c di bng 8l = . d: a(x 1)+ b(y 2) = 0 ax + by a 2b = 0 ( a2 + b2 > 0) V d ct (C) theo dy cung c di 8l = nn khong cch t tm I(2; 1) ca (C) n d bng 3. ( ) 2 2 2 2 2 2 , 3 3 3 a b a b d I d a b a b a b = = = + + 2 0 8 6 0 3 4 a a ab a b = + = = a = 0: chn b = 1 d: y 2 = 0 a = 3 4 b : chn a = 3, b = 4 d: 3x 4 y + 5 = 0. Cu hi tng t: 28. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 27 a) d i qua O, 2 2 ( ) : 2 6 15 0C x y x y+ + = , 8l = . S: : 3 4 0d x y = ; : 0d y = . b) d i qua (5;2)Q , 2 2 ( ) : 4 8 5 0C x y x y+ = , 5 2l = . S: : 3 0d x y = ; : 17 7 71 0d x y = . c) d i qua (9;6)A , 2 2 ( ) : 8 2 0C x y x y+ = , 4 3l = . S: : 2 12d y x= ; 1 21 : 2 2 d y x= + http://www.Luuhuythuong.blogspot.com HT 68. Trong mt phng vi h to ,Oxy cho ng trn (C) : 2 2 2 8 8 0x y x y+ + = . Vit phng trnh ng thng song song vi ng thng : 3 2 0d x y+ = v ct ng trn (C) theo mt dy cung c di 6l = . (C) c tm I(1; 4), bn knh R = 5. PT ng thng c dng: 3 0, 2x y c c+ + = . V ct (C) theo mt dy cung c di bng 6 nn: ( ) 2 3 4 4 10 1 , 4 4 10 13 1 c c d I c + + = = = = + . Vy phng trnh cn tm l: 3 4 10 1 0x y+ + = hoc 3 4 10 1 0x y+ = . Cu hi tng t: a) 2 2 ( ) : ( 3) ( 1) 3C x y + = , : 3 4 2012 0d x y + = , 2 5l = . S: : 3 4 5 0x y + = ; : 3 4 15 0x y = . HT 69. Trong mt phng vi h trc ta ,Oxy cho ng trn 2 2 ( ) :( 4) ( 3) 25C x y+ + = v ng thng : 3 4 10 0x y + = . Lp phng trnh ng thng d bit ( )d v d ct (C) ti A, B sao cho AB = 6. (C) c tm I( 4; 3) v c bn knh R = 5. Gi H l trung im AB, AH = 3. Do d nn PT ca d c dng: 4 3 0x y m+ + = . Ta c: 1( ,( ))d I = IH = 2 2 2 2 5 3 4AI AH = = 2 2 2716 9 4 13 4 3 mm m = + + = = + Vy PT cc ng thng cn tm l: 4 3 27 0x y+ + = v 4 3 13 0x y+ = . HT 70. Trong mt phng vi h to ,Oxy cho ng trn (C): x2 2 2 2 3 0x y y+ = v im M(0; 2). Vit phng trnh ng thng d qua M v ct (C) ti hai im A, B sao cho AB c di ngn nht. (C) c tm I(1; 1) v bn knh R = 5 . IM = 2 5< M nm trong ng trn (C). Gi s d l ng thng qua M v H l hnh chiu ca I trn d. 29. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 28 Ta c: AB = 2AH = 2 2 2 2 2 2 5 2 5 2 3IA IH IH IM = = . Du "=" xy ra H M hay d IM. Vy d l ng thng qua M v c VTPT (1; 1)MI = Phng trnh d: 2 0x y + = . Cu hi tng t: a) Vi (C): 2 2 8 4 16 0x y x y+ = , M(1; 0). S: x: 5 2 5 0d y+ + = HT 71. Trong mt phng vi h to ,Oxy cho ng trn (C) c tm O, bn knh R = 5 v im M(2; 6). Vit phng trnh ng thng d qua M, ct (C) ti 2 im A, B sao cho OAB c din tch ln nht. Tam gic OAB c din tch ln nht OAB vung cn ti O. Khi 5 2 ( , ) 2 d O d = . Gi s phng trnh ng thng d: 2 2 ( 2) ( 6) 0 ( 0)A x B y A B + = + 5 2 ( , ) 2 d O d = A 2 2 2 6 5 2 2 B A B = + A A2 2 47 48 17 0B B+ = 24 5 55 47 24 5 55 47 B A B A = + = + Vi 24 5 55 47 B A = : chn A = 47 B = 24 5 55 d: ( )47( 2) 24 5 55 ( 6) 0x y + = + Vi 24 5 55 47 B A + = : chn A = 47 B = 24 5 55 + d: ( )47( 2) 24 5 55 ( 6) 0x y + + = Cu hi tng t: a) x2 2 ( ) : 4 6 9 0C x y y+ + + = , (1; 8)M . S: x x7 1 0; 17 7 39 0y y+ + = + + = . HT 72. Trong mt phng vi h to ,Oxy cho ng trn (C): 2 2 6 2 6 0x y x y+ + = v im (3;3)A . Lp phng trnh ng thng d qua A v ct (C) ti hai im sao cho khong cch gia hai im bng di cnh hnh vung ni tip ng trn (C). (C) c tm I(3; 1), R = 4. Ta c: A(3 ;3) (C). PT ng thng d c dng: 2 2 ( 3) ( 3) 0, 0a x b y a b + = + 3 3 0ax by a b+ = . Gi s d qua A ct (C) ti hai im A, B AB = 4 2 . Gi I l tm hnh vung. Ta c: 1 1 ( , ) 2 2 ( ) 2 2 d I d AD AB= = = 2 2 3 3 3 2 2 a b a b a b = + 30. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 29 2 2 2 2 4 2 2b a b a b a b = + = = . Chn b = 1 th a = 1 hoc a = 1. Vy phng trnh cc ng thng cn tm l: x 6 0y+ = hoc 0x y = . HT 73. Trong mt phng vi h to ,Oxy cho hai ng trn (C1): 2 2 13x y+ = v (C2): 2 2 ( 6) 25x y + = . Gi A l mt giao im ca (C1) v (C2) vi yA > 0. Vit phng trnh ng thng d i qua A v ct (C1), (C2) theo hai dy cung c di bng nhau. (C1) c tm O(0; 0), bn knh R1 = 13 . (C2) c tm I2(6; 0), bn knh R2 = 5. Giao im A(2; 3). Gi s d: 2 2 ( 2) ( 3) 0 ( 0)a x b y a b + = + . Gi 1 2 2( , ), ( , )d d O d d d I d= = . T gi thit 2 2 2 2 1 1 2 2R d R d = 2 2 2 1 12d d = a a a2 2 2 2 2 2 (6 2 3 ) ( 2 3 ) 12 b b a b a b = + + a2 3 0b b+ = a 0 3 b b = = . Vi b = 0: Chn a = 1 Phng trnh d: 2 0x = . Vi b = 3a: Chn a = 1, b = 3 Phng trnh d: 3 7 0x y + = . HT 74. Trong mt phng vi h ta ,Oxy cho ng thng : 4 0mx y+ = , ng trn (C): 2 2 2 2 2 24 0x y x my m+ + = c tm I. Tm m ng thng ct ng trn (C) ti hai im phn bit A, B sao cho din tch tam gic IAB bng 12. (C) c tm (1; )I m , bn knh R = 5. Gi H l trung im ca dy cung AB. 2 2 4 5 ( , ) 16 16 m m m IH d I m m + = = = + + ; 2 2 2 2 2 (5 ) 20 25 16 16 m AH IA IH m m = = = + + 12IABS = 2 3 ( , ). 12 3 25 48 0 16 3 m d I AH m m m = = + = = HT 75. Trong mat phang toa o ,Oxy cho ng tron 2 2 ( ) : 1C x y+ = , ng thang ( ) : 0d x y m+ + = . Tm m e ( )C cat ( )d tai A va B sao cho dien tch tam giac ABO l n nhat. (C) c tm O(0; 0) , bn knh R = 1. (d) ct (C) ti A, B ( ; ) 1d O d < Khi : 1 1 1 . .sin .sin 2 2 2OABS OAOB AOB AOB= = . Du "=" xy ra 0 90AOB = . Vy AOBS ln nht 0 90AOB = . Khi 1 ( ; ) 2 d I d = 1m = . HT 76. Trong mt phng vi h to ,Oxy cho ng thng ( )d : 2 1 2 0x my+ + = v ng trn c phng 31. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 30 trnh 2 2 ( ) : 2 4 4 0C x y x y+ + = . Gi I l tm ng trn ( )C . Tm m sao cho ( )d ct ( )C ti hai im phn bit A v B. Vi gi tr no ca m th din tch tam gic IAB ln nht v tnh gi tr . ( )C c tm I (1; 2) v bn knh R = 3. (d) ct ( )C ti 2 im phn bit A, B ( , )d I d R < 22 2 1 2 3 2m m + < + 2 2 21 4 4 18 9 5 4 17 0m m m m m m R + < + + + > Ta c: 1 1 9 . sin . 2 2 2 S IAIB AIB IAIB IAB = = Vy: S IAB ln nht l 9 2 khi 0 90AIB = AB = 2 3 2R = 3 2 ( , ) 2 d I d = 3 2 21 2 2 2 m m = + 22 16 32 0m m + + = 4m = Cu hi tng t: a) Vi : 2 3 0d x my m+ + = , 2 2 ( ) : 4 4 6 0C x y x y+ + + + = . S: 8 0 15 m m= = HT 77. Trong mt phng vi h to ,Oxy cho ng trn x2 2 ( ) : 4 6 9 0C x y y+ + + = v im (1; 8)M . Vit phng trnh ng thng d i qua M, ct (C) ti hai im A, B phn bit sao cho tam gic ABI c din tch ln nht, vi I l tm ca ng trn (C). (C) c tm ( 2;3)I , bn knh 2R = . PT ng thng d qua (1; 8)M c dng: : 8 0d ax by a b+ + = ( 2 2 0a b+ ). 1 . .sin 2 sin 2IABS IAIB AIB AIB = = . Do : IABS ln nht 0 90AIB = 2 ( , ) 2 2 d I d IA= = a 2 2 11 3 2 b a b = + a a2 2 7 66 118 0b b + = a 7 7 17 a b b = = . + Vi 1 7b a= = x: 7 1 0d y+ + = + Vi 7 17b a= = x: 17 7 39 0d y+ + = HT 78. Trong mt phng vi h ta ,Oxy cho ng trn (C): 2 2 4 4 6 0x y x y+ + + + = v ng thng : 2 3 0x my m+ + = vi m l tham s thc. Gi I l tm ca ng trn (C). Tm m ct (C) ti 2 im phn bit A v B sao cho din tch IAB ln nht. (C) c tm l I (2; 2); R = 2 . Gi s ct (C) ti hai im phn bit A, B. K ng cao IH ca IAB, ta c: SABC = 1 . .sin 2IABS IAIB AIB= = sinAIB 32. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 31 Do IABS ln nht sin AIB = 1 AIB vung ti I IH = 1 2 IA = (tha IH < R) 2 1 4 1 1 m m = + 15m2 8m = 0 m = 0 hay m = 8 15 Cu hi tng t: a) Vi 2 2 ( ) : 2 4 4 0C x y x y+ + = , : 2 1 2 0x my + + = . S: 4m = . b) Vi 2 2 ( ) : 2 4 5 0C x y x y+ = , : 2 0x my + = . S: 2m = HT 79. Trong mt phng vi h ta ,Oxy cho ng thng d: 5 2 0x y = v ng trn (C): 2 2 2 4 8 0x y x y+ + = . Xc nh ta cc giao im A, B ca ng trn (C) v ng thng d (cho bit im A c honh dng). Tm ta C thuc ng trn (C) sao cho tam gic ABC vung B. Ta giao im A, B l nghim ca h phng trnh 2 2 0; 22 4 8 0 1; 35 2 0 y xx y x y y xx y = =+ + = = = = . V 0Ax > nn ta c A(2;0), B(3;1). V 0 90ABC = nn AC l ng knh ng trn, tc im C i xng vi im A qua tm I ca ng trn. Tm I(1;2), suy ra C(4;4). HT 80. Trong mt phng vi h ta ,Oxy cho ng trn (C ): 2 2 2 4 8 0x y x y+ + = v ng thng ( ): 2 3 1 0x y = . Chng minh rng ( ) lun ct (C ) ti hai im phn bit A, B . Tm to im M trn ng trn ( C ) sao cho din tch tam gic ABM ln nht. (C) c tm I(1; 2), bn knh R = 13 . 9 ( , ) 13 d I R = < ng thng ( ) ct (C) ti hai im A, B phn bit. Gi M l im nm trn (C), ta c 1 . ( , ) 2ABMS AB d M = . Trong AB khng i nn ABMS ln nht ( , )d M ln nht. Gi d l ng thng i qua tm I v vung gc vi ( ). PT ng thng d l 3 2 1 0x y+ = . Gi P, Q l giao im ca ng thng d vi ng trn (C). To P, Q l nghim ca h phng trnh: 2 2 2 4 8 0 3 2 1 0 x y x y x y + + = + = 1, 1 3, 5 x y x y = = = = P(1; 1); Q(3; 5) Ta c 4 ( , ) 13 d P = ; 22 ( , ) 13 d Q = . Nh vy ( , )d M ln nht M trng vi Q. Vy ta im M(3; 5). HT 81. Trong mt phng vi h to ,Oxy cho ng trn (C): x2 2 2 4 5 0x y y+ = v A(0; 1) (C). Tm to cc im B, C thuc ng trn (C) sao cho ABC u. 33. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 32 (C) c tm I(1;2) v R= 10 . Gi H l trung im BC. Suy ra 2.AI IH= 3 7 ; 2 2 H ABC u I l trng tm. Phng trnh (BC): 3 12 0x y+ = V B, C (C) nn ta ca B, C l cc nghim ca h phng trnh: 2 2 2 2 2 4 5 0 2 4 5 0 3 12 0 12 3 x y x y x y x y x y x y + = + = + = = Gii h PT trn ta c: 7 3 3 3 3 7 3 3 3 3 ; ; ; 2 2 2 2 B C + + hoc ngc li. HT 82. Trong mt phng vi h to ,Oxy cho ng trn (C): 2 2 ( 3) ( 4) 35x y + = v im A(5; 5). Tm trn (C) hai im B, C sao cho tam gic ABC vung cn ti A. (C) c tm I(3; 4). Ta c: AB AC IB IC = = AI l ng trung trc ca BC. ABC vung cn ti A nn AI cng l phn gic ca BAC . Do AB v AC hp vi AI mt gc 0 45 . Gi d l ng thng qua A v hp vi AI mt gc 0 45 . Khi B, C l giao im ca d vi (C) v AB = AC. V (2;1)IA = (1; 1), (1; 1) nn d khng cng phng vi cc trc to VTCP ca d c hai thnh phn u khc 0. Gi (1; )u a= l VTCP ca d. Ta c: ( ) 2 2 2 2 2 2 cos , 21 2 1 5 1 a a IA u a a + + = = = + + + 2 2 2 5 1a a+ = + 3 1 3 a a = = + Vi a = 3, th (1;3)u = Phng trnh ng thng d: 5 5 3 x t y t = + = + . Ta tm c cc giao im ca d v (C) l: 9 13 7 3 13 9 13 7 3 13 ; , ; 2 2 2 2 + + + Vi a = 1 3 , th 1 1; 3 u = Phng trnh ng thng d: 5 1 5 3 x t y t = + = . Ta tm c cc giao im ca d v (C) l: 7 3 13 11 13 7 3 13 11 13 ; , ; 2 2 2 2 + + +V AB = AC nn ta c hai cp im cn tm l: 7 3 13 11 13 9 13 7 3 13 ; , ; 2 2 2 2 + + + v 7 3 13 11 13 9 13 7 3 13 ; , ; 2 2 2 2 + 34. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S N BN Page 33 HT 83. Trong mt phng to ,Oxy cho ng trn (C): 2 2 4x y+ = v cc im 8 1; 3 A , (3;0)B . Tm to im M thuc (C) sao cho tam gic MAB c din tch bng 20 3 . 64 10 4 ; : 4 3 12 0 9 3 AB AB x y= + = = . Gi M(x;y) v ( , )h d M AB= . Ta c: 4 3 8 04 3 121 20 . 4 4 4 3 32 02 3 5 x yx y h AB h x y + = = = = = + 2 2 4 3 8 0 14 48 ( 2;0); ; 25 754 x y M M x y + = + = + 2 2 4 3 32 0 4 x y x y = + = (v nghim) HT 84. Trong mt phng to ,Oxy cho ng trn x2 2 ( ) : 2 6 9 0C x y y+ + + = v ng thng x: 3 4 5 0d y + = . Tm nhng im M (C) v N d sao cho MN c di nh nht. (C) c tm ( 1;3)I , bn knh 1R = ( , ) 2d I d R= > ( )d C = . Gi l ng thng qua I v vung gc vi d x( ) : 4 3 5 0y + = . Gi 0 0 1 7 ; 5 5 N d N = . Gi 1 2,M M l cc giao im ca v (C) 1 2 2 11 8 19 ; , ; 5 5 5 5 M M MN ngn nht khi 1 0,M M N N . Vy cc im cn tm: 2 11 ; ( ) 5 5 M C , 1 7 ; 5 5 N d . 35. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 34 PHN III CC BI TON LIN QUAN N TAM GIC Ton b ti li u luy n thi i h c mn ton c a th y Lu Huy Th ng: http://www.Luuhuythuong.blogspot.com HT 85.Trong mt phng vi h to Oxy, cho tam gic ABC c 3 nh ( 5;3), (2; 1), ( 1;3)A B C . a) Vit phng trnh ba cnh ca tam gic. b) Vit phng trnh ng trung tuyn AM c) Vit phng trnh ng cao BH d) Vit phng trnh ng trung trc d ca cnh AC. e) Vit phng trnh ng phn gic trong nh C. Gii a) Cnh ( 5;3) (7; 4) qua A AB vtcp AB = 5 3 : 7 4 x y AB + = Cnh ( 5;3) : (4;0) qua A AC vtcp AC = 5 4 : 3 x t AC y = + = Cnh (2; 1) : ( 3;4) qua B BC vtcp BC = 2 1 : 3 4 x x BC + = b) Ta c, M l trung im ca BC 1 ( ;1) 2 M Phng trnh ng trung tuyn ( 5;3) : 11 ( ; 2) 2 qua A AM vtcp AM = 5 3 : 11 2 2 x x AM + = c) ng cao (2; 1) 2 4 : : 1(4;0) qua B x t BH BH yvtpt AC = + == d) Gi N l trung im ca AC ( 3;3)N ng trung trc ca AC : ( 3;3) 3 4 : : 3(4;0) qua N x t d d yvtpt AC = + == e) Ta c : : 4 3 5 0BC x y+ = ; : 3 0AC y = Phng trnh ng phn gic gc to bi BC v AC l : 4 3 5 3 5 1 x y y+ = 4 3 5 5 15 4 3 5 5 15 x y y x y y + = + = + 1 2 2 5 0 ( ) 2 5 0 ( ) x y l x y l + = + = Xt v tr tng i ca A v B so vi 1l Ta c : 2.( 5) 3 5 8At = + = ; 2.2 1 5 10Bt = + + = 36. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 35 . 80 0A Bt t = < Vy, A v B nm khc pha so vi 1l nn 1l l ng phn gic trong nh C. HT 86.Trong mt phng vi h to Oxy, cho tam gic ABC c (2;2)A v phng trnh hai ng cao k t B v C ln lt l: 1 2: 9 3 4 0; : 2 0d x y d x y = + = . Vit phng trnh cc cnh ca tam gic ABC. Gii Lp phng trnh AC : Ta c : 1 1: 3 0AC d AC x y c + + = AC qua 1(2;2) 8 : 3 8 0A c AC x y = + = Lp phng trnh AB : Ta c : 2 2: 0AB d AB x y c + = AB qua 2(2;2) 0 : 0A c AB x y = = Lp phng trnh BC : Ta im B l nghim ca h phng trnh : 2 0 2 23 ; 9 3 4 0 2 3 3 3 xx y B x y y = = = = Ta im C l nghim ca h phng trnh : 3 8 0 1 ( 1;3) 2 0 3 x y x C x y y + = = + = = Vy, ( 1;3) : : 7 5 8 05 7 ( ; ) 7 3 quaC BC BC x y vtcp BC + = = http://www.Luuhuythuong.blogspot.com HT 87.Trong mt phng vi h to Oxy, cho tam gic ABC c phng trnh hai ng cao ln lt l: : 4 1 0AH x y = v : 3 0BK x y + = , trng tm tam gic (1;2)G . Vit phng trnh cc cnh ca tam gic. Gii Ta im ( ;4 1); ( ; 3)A a a B b b + H thc trng tm trong tam gic : 3 3 A B C G A B C G x x x x y y y y + + = + + = 1 3 3 4 44 1 3 2 3 C C C a b x x a b y a ba b + + = = = + + = (3 ;4 4 )C a b a b (3 2 ;5 8 )AC a b a b= ; (3 2 ;1 4 2 )BC a b a b= 37. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 36 Ta c: . 0 3 2 5 8 0 3 2 4 16 8 0. 0 BK AH AC BK AC u a b a b BC AH a b a bBC u = + = + == 5 4 1 17 10 7 1 a b a a b b + = = + = = Vy, (1;3), ( 1;2), (3;1)A B C Phng trnh cc cnh (hc sinh t vit) : 2 5 0; : 4 0, : 4 7 0AB x y AC x y BC x y + = + = + = HT 88.Trong mt phng vi h to Oxy, cho tam gic ABC c nh A(1; 3) v hai ng trung tuyn ca n c phng trnh l: 2 1 0x y + = v 1 0y = . Hy vit phng trnh cc cnh ca ABC. Gii Thay ta im A vo phng trnh hai ng trung tuyn ta thy khng tha mn. Khng mt tnh tng qut, t trung tuyn : 2 1 0BM x y + = , trung tuyn : 1 0CN y = Ta trng tm G l nghim ca h phng trnh: 2 1 0 1 0 x y y + = = 1 (1;1) 1 x G y = = Ta c, ta (2 1; ); ( ;1)B b b C c H thc trng tm tam gic: 1 2 1 1 1 3 3 3 1 5 1 33 A B C G A B C G x x x b c x b y y y b c y + + + + = = = + + + + = == Vy, ( 3; 1), (5;1)B C Phng trnh ba cnh (hc sinh t vit) (AC): x + 2y 7 = 0; (AB): x y + 2 = 0; (BC): x 4y 1 = 0. HT 89.Trong mt phng vi h to Oxy, cho tam gic ABC c (2;1)A . ng cao BH c phng trnh 3 7 0x y = . ng trung tuyn CM c phng trnh 1 0x y+ + = . Xc nh to cc nh B, C. Tnh din tch tam gic ABC. Gii AC qua A v vung gc vi ng cao BH ( ) : 3 7 0AC x y = . To im C l nghim ca h: 3 7 0 1 0 x y x y = + + = (4; 5)C . Trung im M ca AB c: 2 1 ; 2 2 B B M M x y x y + + = = . ( )M CM 2 1 1 0 2 2 B Bx y+ + + + = . 38. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 37 To im B l nghim ca h: 3 7 0 2 1 1 0 2 2 B B x y x y = + + + + = ( 2; 3)B . To im H l nghim ca h: x 3 7 0 3 7 0 x y y = + = 14 7 ; 5 5 H . 8 10 ; 2 10 5 BH AC= = 1 1 8 10 . .2 10. 16 2 2 5ABCS AC BH = = = (vdt). HT 90.Trong mt phng vi h ta Oxy, cho ABC c ta nh B(3; 5) , phng trnh ng cao h t nh A v ng trung tuyn h t nh C ln lt l 1d : 2x 5y + 3 = 0 v 2d : x + y 5 = 0. Tm ta cc nh A v C ca tam gic ABC. Gii Gi M l trung im AB th M 2d nn ( ;5 )M a a . nh A 1d nn 5 3 ; 2 b A b . M l trung im AB: 2 2 A B M A B M x x x y y y + = + = 4 5 3 2 2 5 1 a b a a b b = = + = = A(1; 1). Phng trnh BC: x5 2 25 0y+ = ; 2C d BC= C(5; 0). HT 91.Trong mt phng vi h to Oxy, cho tam gic ABC c (4; 2)A , phng trnh ng cao k t C v ng trung trc ca BC ln lt l: 2 0x y + = , x3 4 2 0y+ = . Tm to cc nh B v C. Gii ng thng AB qua A v vung gc vi ng cao CH ( ) : 2 0AB x y + = . Gi ( ;2 ) ( )B b b AB , ( ; 2) ( )C c c CH+ Trung im M ca BC: 4 ; 2 2 b c b c M + + . V M thuc trung trc ca BC nn: 3( ) 4(4 ) 4 0b c b c+ + + = 7 12 0b c + + = (1) ( ; )BC c b c b= + l 1 VTPT ca trung trc BC nn 4( ) 3( )c b c b = + 7c b= (2) T (1) v (2) 7 1 , 4 4 c b= = . Vy 1 9 7 1 ; , ; 4 4 4 4 B C . HT 92.Trong mt phng vi h to Oxy, cho tam gic ABC bit A(5; 2). Phng trnh ng trung trc cnh BC, ng trung tuyn CC ln lt l x + y 6 = 0 v 2x y + 3 = 0. Tm ta cc nh ca tam gic ABC. Gii Gi ( ; 2 3)C c c + v ( ;6 )I m m l trung im ca BC. Suy ra: (2 ; 9 2 2 )B m c m c . V C l trung im ca AB nn: 2 5 11 2 2 ' ; ' 2 2 m c m c C CC + 39. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 38 nn 2 5 11 2 2 5 2 3 0 2 2 6 m c m c m + + = = 5 41 ; 6 6 I . HT 93. Phng trnh BC: 3 3 23 0x y + = 14 37 ; 3 3 C 19 4 ; 3 3 B .Trong mt phng vi h ta Oxy , cho tam gic ABC c nh A(3; 4). Phng trnh ng trung trc cnh BC, ng trung tuyn xut pht t C ln lt l 1 : 1 0d x y+ = v 2 : 3 9 0d x y = . Tm ta cc nh B, C ca tam gic ABC. Gii Gi 2( ;3 9)C c c d v M l trung im ca BC 1( ;1 )M m m d . (2 ;11 2 3 )B m c m c . Gi I l trung im ca AB, ta c 2 3 7 2 3 ; 2 2 m c m c I + . V I 2( )d nn 2 3 7 2 3 3. 9 0 2 2 m c m c + = 2m = (2; 1)M Phng trnh BC: 3 0x y = . 2 (3;0) (1; 2)C BC d C B= . HT 94.Trong mt phng vi h to Oxy, cho tam gic ABC c A(3; 5); B(4; 3), ng phn gic trong v t C l : 2 8 0d x y+ = . Lp phng trnh ng trn ngoi tip tam gic ABC. Gii Gi E l im i xng ca A qua d E BC. Tm c (1;1)E PT ng thng BC: x4 3 1 0y+ + = . C d BC= ( 2;5)C . Phng trnh ng trn (ABC) c dng: 2 2 2 2 2 2 0; 0x y ax by c a b c+ + = + > Ta c A, B, C (ABC) 4 10 29 1 5 99 6 10 34 ; ; 2 8 4 8 6 25 a b c a b c a b c a b c + = + = = = = + + = Vy phng trnh ng trn l: 2 2 5 99 0 4 4 x y x y+ = . http://www.Luuhuythuong.blogspot.com HT 95.Trong mt phng vi h to Oxy, cho ABC bit: B(2; 1), ng cao qua A c phng trnh d1: 3 4 27 0x y + = , phn gic trong gc C c phng trnh d2: 2 5 0x y+ = . Tm to im A. Gii Phng trnh BC: 2 1 3 4 x y + = To im ( 1;3)C + Gi B l im i xng ca B qua d2, I l giao im ca BB v d2. 40. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 39 phng trnh BB: 2 1 1 2 x y + = 2 5 0x y = + To im I l nghim ca h: 2 5 0 3 (3;1) 2 5 0 1 x y x I x y y = = + = = + V I l trung im BB nn: ' ' 2 4 (4;3) 2 3 B I B B I B x x x B y y y = = = = + ng AC qua C v B nn c phng trnh: y 3 =0. + To im A l nghim ca h: 3 0 5 ( 5;3) 3 4 27 0 3 y x A x y y = = + = = HT 96.Trong mt phng vi h to Oxy, cho tam gic ABC c phng trnh ng phn gic trong gc A l (d1): 2 0x y+ + = , phng trnh ng cao v t B l (d2): 2 1 0x y + = , cnh AB i qua M(1; 1). Tm phng trnh cnh AC. Gii Gi N l im i xng ca M qua (d1) N AC . ( 1, 1)N NMN x y= + Ta c: 1 / / (1; 1)dMN n = 1( 1) 1( 1) 0 2 (1)N N N Nx y x y + = = Ta trung im I ca MN: 1 1 (1 ), ( 1 ) 2 2I N I Nx x y y= = + 1 1 1 ( ) (1 ) ( 1 ) 2 0 2 2N NI d x y + + + = 4 0 (2)N Nx y + + = Gii h (1) v (2) ta c N(1; 3) Phng trnh cnh AC vung gc vi (d2) c dng: x + 2y + C = 0. ( ) 1 2.( 3) 0 7.N AC C C + + = = Vy, phng trnh cnh AC: x + 2y + 7 = 0. HT 97.Trong mt phng vi h to Oxy, cho tam gic ABC vi A(1; 2), ng cao : 1 0CH x y + = , phn gic trong : 2 5 0BN x y+ + = . Tm to cc nh B, C v tnh din tch tam gic ABC. Gii Do AB CH nn phng trnh AB: 1 0x y+ + = . + B = AB BN To im B l nghim ca h: 2 5 0 1 0 x y x y + + = + + = 4 3 x y = = ( 4;3)B . + Ly A i xng vi A qua BN th 'A BC . Phng trnh ng thng (d) qua A v vung gc vi BN l (d): 2 5 0x y = . 41. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 40 Gi ( )I d BN= . Gii h: 2 5 0 2 5 0 x y x y + + = = . Suy ra: I(1; 3) '( 3; 4)A + Phng trnh BC: 7 25 0x y+ + = . Gii h: : 7 25 0 : 1 0 BC x y CH x y + + = + = 13 9 ; 4 4 C . + 2 2 13 9 450 4 3 4 4 4 BC = + + + = , 2 2 7.1 1( 2) 25 ( ; ) 3 2 7 1 d A BC + + = = + . Suy ra: 1 1 450 45 ( ; ). .3 2. . 2 2 4 4ABCS d A BC BC= = = HT 98.Trong mt phng vi h to Oxy, cho tam gic ABC c nh (2; 1)B , ng cao xut pht t A v ng phn gic trong gc C ln lt l x1 : 3 4 27 0d y + = , 2 : 2 5 0d x y+ = . Vit phng trnh cc cnh ca tam gic ABC. Gii ng thng BC qua B v vung gc vi 1d x( ) : 4 3 5 0BC y+ + = . To nh C l nghim ca h: x4 3 5 0 2 5 0 y x y + + = + = ( 1;3)C . Gi B l im i xng ca B qua 2d (4;3)B v ( )B AC . ng thng AC i qua C v B ( ) : 3 0AC y = . To nh A l nghim ca h: x 3 0 3 4 27 0 y y = + = ( 5;3)A . ng thng AB qua A v B x( ) : 4 7 1 0AB y+ = . Vy: x( ) : 4 7 1 0AB y+ = , x( ) : 4 3 5 0BC y+ + = ,( ) : 3 0AC y = . http://www.Luuhuythuong.blogspot.com HT 99.Trong mt phng vi h to Oxy, cho ABC c nh A(1;2), phng trnh ng trung tuyn BM: 2 1 0x y+ + = v phn gic trong CD: 1 0x y+ = . Vit phng trnh ng thng BC. Gii im : 1 0 ( ;1 )C CD x y C t t + = . Suy ra trung im M ca AC l 1 3 ; 2 2 t t M + . T A(1;2), k : 1 0AK CD x y + = ti I (im K BC ). Suy ra : ( 1) ( 2) 0 1 0AK x y x y = + = . 42. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 41 Ta im I tha h: ( ) 1 0 0;1 1 0 x y I x y + = + = Tam gic ACK cn ti C nn I l trung im ca AK ta ca ( 1;0)K . ng thng BC i qua C, K nn c phng trnh: 1 4 3 4 0 7 1 8 x y x y + = + + = + HT 100.Trong mt phng vi h to Oxy, cho tam gic ABC c nh C(4; 3). Bit phng trnh ng phn gic trong (AD): 2 5 0x y+ = , ng trung tuyn (AM): x4 13 10 0y+ = . Tm to nh B. Gii Ta c A = AD AM A(9; 2). Gi C l im i xng ca C qua AD C AB. Ta tm c: C(2; 1). Suy ra phng trnh (AB): 9 2 2 9 1 2 x y + = + 7 5 0x y+ + = . Vit phng trnh ng thng Cx // AB (Cx): 7 25 0x y+ = HT 101.Trong mt phng vi h to Oxy, cho ABC , vi nh A(1; 3) phng trnh ng phn gic trong BD: 2 0x y+ = v phng trnh ng trung tuyn CE: 8 7 0x y+ = . Tm to cc nh B, C. Gii Gi E l trung im ca AB. Gi s D( ;2 )B b b B 1 1 ; 2 2 b b E CE + + 3b = ( 3;5)B . Gi A l im i xng ca A qua BD A BC. Tm c A(5; 1) Phng trnh BC: 2 7 0x y+ = ; 8 7 0 : (7;0) 2 7 0 x y C CE BC C x y + == + = . HT 102.Trong mt phng vi h trc ta Oxy cho tam gic ABC vi (1; 2)B ng cao : 3 0AH x y + = . Tm ta cc nh A, C ca tam gic ABC bit C thuc ng thng :2 1 0d x y+ = v din tch tam gic ABC bng 1. Gii Phng trnh : 1 0BC x y+ + = . C = BC d (2; 3)C . Gi 0 0 0 0( ; ) 3 0A x y AH x y + = (1); 0 0 1 2, ( , ) 2 x y BC AH d A BC + + = = = 0 0 0 0 0 0 1 1 2 (2)1 1 . 1 . . 2 1 1 2 (3)2 2 2 ABC x y x y S AH BC x y + + + + == = = + + = 43. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 42 T (1) v (2) 0 0 1 ( 1;2) 2 x A y = = . T (1) v (3) 0 0 3 ( 3;0) 0 x A y = = HT 103.Trong mt phng vi h to Oxy, cho tam gic ABC c nh ( 1; 3)A , trng tm (4; 2)G , trung trc ca AB l x: 3 2 4 0d y+ = . Vit phng trnh ng trn ngoi tip tam gic ABC. Gii Gi M l trung im ca BC 3 2 AM AG= 13 3 ; 2 2 M . AB d AB nhn (2; 3)du = lm VTPT Phng trnh x: 2 3 7 0AB y = . Gi N l trung im ca AB N = AB d (2; 1)N (5;1)B (8; 4)C . PT ng trn (C) ngoi tip ABC c dng: ax2 2 2 2 0x y by c+ + + + = ( 2 2 0a b c+ > ). Khi ta c h: a a a 2 6 10 10 2 26 16 8 80 b c b c b c + = + + = + = 74 21 23 7 8 3 a b c = = = . Vy: 2 2 148 46 8 ( ) : 0 21 7 3 C x y x y+ + + = HT 104.Trong mt phng vi h to Oxy, cho tam gic ABC c phn gic trong AD v ng cao CH ln lt c phng trnh 2 0x y+ = , 2 5 0x y + = . im (3;0)M thuc on AC tho mn A2AB M= . Xc nh to cc nh A, B, C ca tam gic ABC. Gii Gi E l im i xng ca M qua AD (2; 1)E . ng thng AB qua E v vung gc vi CH x( ) : 2 3 0AB y+ = . To im A l nghim ca h: x2 3 0 2 0 y x y + = + = (1;1)A PT ( ) : 2 3 0AM x y+ = Do A2AB M= nn E l trung im ca AB (3; 3)B . To im C l nghim ca h: 2 3 0 2 5 0 x y x y + = + = ( 1;2)C Vy: (1;1)A , (3; 3)B , ( 1;2)C . Cu hi tng t: a) D( ) : 0A x y = , x( ) : 2 3 0CH y+ + = , (0; 1)M . S: (1;1)A ; ( 3; 1)B ; 1 ; 2 2 C 44. GV. Lu Huy Th ng 0968.393.899 B H C V B - CHUYN C N S N B N Page 43 HT 105.Trong mt phng vi h to Oxy, cho tam gic ABC cn ti A, ng thng BC c phng trnh 2 2 0x y+ = . ng cao k t B c phng trnh 4 0x y + = , im ( 1;0)M thuc ng cao k t C. Xc nh to cc nh ca tam gic ABC. Gii To nh B l nghim ca h: 2 2 0 4 0 x y x y + = + = ( 2;2)B . Gi d l ng thng qua M v song song vi BC : 2 1 0d x y+ + = . Gi N l giao im ca d vi ng cao k t B To ca N l nghim ca h: 4 0 2 1 0 x y x y + = + + = ( 3;1)N . Gi I l trung im ca MN 1 2; 2 I . Gi E l trung im ca BC IE l ng trung trc ca BC x: 4 2 9 0IE y + = . To im E l nghim ca h: x 2 2 0 4 2 9 0 x y y + = + = 7 17 ; 5 10 E 4 7 ; 5 5 C . ng thng CA qua C v vung gc vi BN 3 : 0 5 CA x y+ = . To nh A l nghim ca h: x4 2 9 0 3 0 5 y x y + = + = 13 19 ; 10 10 A . Vy: 13 19 ; 10 10 A , ( 2;2)B , 4 7 ; 5 5 C . HT 106.Trong mt phng vi h to Oxy, cho tam gic ABC c nh A thuc ng thng d: 4 2 0x y = , cnh BC song song vi d, phng trnh ng cao BH: 3 0x y+ + = v trung im ca cnh AC l M(1; 1). Tm to cc nh A, B, C. Gii Ta c AC vung gc vi BH v i qua M(1; 1) nn c phng trnh: y x= . To nh A l nghim ca h : 2 4 2 0 2 23 ; 2 3 3 3 xx y A y x y = = = = V M l trung im ca AC nn 8 8 ; 3 3 C V BC i qua C v song song vi d nn BC c phng trnh: 2 4 x y = +