Cac Bai Toan Duong Thang Oxy 1997-2012

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Bai 1. Cao ng S Phm Nh Tr Mu Gio TW1 nm 2000

Trong mt phng ta Oxy, cho ng thng : 2 3 3 0x y . Vit phng trnh ng thng i qua 5;13M v vung gc vi ng thng . S: :3 2 11 0d x y .

Bai 2. Cao ng S Phm H Ni nm 1997

Trong mt phng ta Oxy, cho ABC vi 1; 1 , 2;1 , 3;5A B C .

1/ Vit phng trnh ng vung gc AH k t A n trung tuyn BK ca ABC.

2/ Tnh din tch ABK.

S: 1/ : 4 3 0AH x y . 2/ 11 ABKS vdt .

Bai 3. Cao ng K Ngh Tp. H Ch Minh nm 1998

Trong mt phng ta Oxy, cho hai ng thng: 1 : 4 3 12 0x y v 2 : 4 3 12 0x y .

1/ Xc nh nh ca tam gic c ba cnh thuc 1 2, v trc Oy .

2/ Tm ta tm v bn knh ng trn ni tip tam gic ni trn.

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S: 1/

1

2

1 2

0; 4

0;4

3;0

A Oy

B Oy

C

. 2/

4 ;0

3

4: ;

3

Tm I

Bk R d I AB

.

Bai 4. Cao ng S Phm H Ni khi A nm 1999

Trong mt phng ta Oxy, cho ABC, cnh BC, cc ng cao BI, CK c phng trnh ln lt l 7 5 8 0,x y 9 3 4 0,x y 2 0x y . Vit phng trnh cc cnh

AB, AC v ng cao AH.

S: : 0, : 3 8 0, :5 7 4 0AB x y AC x y AH x y .

Bai 5. Cao ng Cng Nghip Tp. H Ch Minh nm 2000

Trong mt phng ta Oxy, cho ABC c cc ng cao : 1 0BH x y , : 3 1 0CK x y v cnh :5 5 0BC x y . Vit phng trnh ca cc cnh cn li ca tam gic v ng cao AL ?

S: : 3 1 0, : 3 0, : 5 3 0AB x y AC x y AL x y .

Bai 6. Cao ng Kim St Pha Bc nm 2000

Trong mt phng ta Oxy, cho ABC c 1;3A v hai trung tuyn l 2 1 0x y v 1 0y . Vit phng trnh cc cnh ca tam gic ?

S: : 2 0, : 2 3 0, : 4 1 0AB x y AC x y BC x y .

Bai 7. Cao ng S Phm Nh Tr Mu Gio TWI nm 2001

Trong mt phng ta Oxy, cho cc im 1;2 , 1;2A B v ng thng d c phng trnh : 2 1 0d x y . Hy tm ta ca im C thuc ng thng d sao cho ba im A, B, C to thnh tam gic v tha mn mt trong cc iu kin sau

1/ CA CB . 2/ AB AC .

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S: 1/ 1

0;2

C

. 2/ 1 2

3;2 ;5 5

C C

.

Bai 8. Cao ng S Phm Vnh Phc khi A nm 2002

Trong mt phng ta Oxy, cho ABC v im 1;1M l trung im ca AB. Hai cnh AC v BC theo th t nm trn hai ng thng 2 2 0x y v 3 3 0x y .

1/ Xc nh ta ba nh A, B, C ca ABC v vit phng trnh ng cao CH.

2/ Tnh din tch ABC.

S: 1/ 3 4

1;0 , 3;2 , ;5 5

A B C

v :10 5 2 0CH x y . 2/ 6

5

ABCS vdt

.

Bai 9. Cao ng Nng Lm nm 2003

Trong mt phng vi h trc ta Oxy, cho hai ng thng 1 0x y v

3 5 0x y . Hy tm din tch hnh bnh hnh c hai cnh nm trn hai ng thng

cho, mt nh l giao im ca hai ng thng v giao im ca hai ng cho l

3;3I .

S: 55 ABCDS vdt .

Bai 10. Cao ng S Phm Ph Th khi A nm 2003

Trong mt phng vi h trc ta cac Oxy cho tam gic ABC c nh 2; 3 ,A

3; 2B v din tch tam gic ABC bng 3

2. Bit trng tm G ca ABC thuc ng

thng :3 8 0d x y . Tm ta im C.

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S: 1; 1 4;8C C .

Bai 11. Cao ng khi D, M nm 2004 i hc Hng Vng

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC bit nh

3;9A v phng trnh cc ng trung tuyn BM, CN ln lt l 3 4 9 0,x y 6 0y . Vit phng trnh ng trung tuyn AD ca tam gic cho.

S: :3 2 27 0AD x y .

Bai 12. Cao ng iu Dng chnh quy nm 2004 i hc iu dng

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC c nh 0;1A v hai ng thng cha cc ng cao v t B v C c phng trnh tng ng l 2 1 0x y v 3 1 0x y . Tnh din tch ABC.

S: 14 ABCS vdt .

Bai 13. Cao ng khi A nm 2004

Cho tam gic ABC c 6; 3 , 4;3 , 9;2A B C . 1/ Vit phng trnh cc cnh ca ABC.

2/ Vit phng trnh ng phn gic trong ca gc A ca tam gic ABC.

3/ Tm im M trn cnh AB v tm im N trn cnh AC sao cho MN // BC v AM CN .

S: 1/

:3 15 0

: 3 3 0

: 13 35 0

AB x y

AC x y

BC x y

. 2/ : 3A

d y x . 3/ 32 9 33 4

; , ;7 7 7 7

M N

.

Bai 14. Cao ng S Phm Hi Phng nm 2004

Trong mt phng vi h trc ta Oxy, cho hai ng thng 1: 1 0,x y

2: 2 1 0x y v im 2;1P .

1/ Vit phng trnh ng thng i qua im P v giao im I ca hai ng thng 1 v 2.

2/ Vit phng trnh ng thng i qua im P v ct hai ng thng 1, 2 ln lt ti hai im A, B sao cho P l trung im AB.

S: 1/ 1 0y . 2/ : 4 7 0d AB x y (c th gii theo 3 cch).

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Bai 15. Cao ng S Phm Kom Tum nm 2004

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho hai im 1;2A v 3;4B . Tm im C trn ng thng : 2 1 0d x y sao cho ABC vung C.

S: 3 4

3;2 ;5 5

C C

.

Bai 16. Cao ng Kinh T K Thut Cng Nghip I khi B nm 2004

Trong mt phng vi h trc ta Oxy cho ng thng : 2 3 1 0d x y v im

1;1M . Vit phng trnh ca cc ng thng i qua im M v to vi ng thng d mt gc 045 .

S: 5 4 0x y . C th gii theo hai cch.

Bai 17. Cao ng Kinh T K Thut Cng Nghip I khi A nm 2004

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho hai im 3; 1A v 3;5B . Hy vit phng trnh ng thng i qua im 2;3I v cch u hai im A,

B.

S: 2 0 5 13 0x x y .

Bai 18. Cao ng Mu Gio TW 1 nm 2004

Trong mt phng vi h trc ta Decac Oxy, xt ABC vi : 2 7 0AB x y , cc

ng trung tuyn k t A, B ln lt c phng trnh 5 0x y v 2 11 0x y .

Hy tnh din tch ca ABC v lp phng trnh hai ng thng AC v BC.

S: 45

2

ABCS vdt

v :16 13 68 0, :17 11 106 0AC x y BC x y .

Bai 19. Cao ng khi T M trng i hc Hng Vng nm 2004

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC bit nh

3;9A v phng trnh cc ng trung tuyn BM, CN ln lt l :3 4 9 0x y v 6 0y . Vit phng trnh ng trung tuyn AD.

S: :3 2 27 0AD x y .

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Bai 20. Cao ng Cng Nghip IV nm 2004

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC vung ti A vi

3;0 , 7;0 ,B C bn knh ng trn ni tip 2 10 5r . Tm ta tm I ca ng trn ni tip ABC, bit im I c honh dng.

S: 2 10; 2 10 5 2 10; 2 10 5I I . + Cao ng Ti Chnh K Ton nm 2004

Trong mt phng vi h trc ta Oxy, cho ba im 2;1 , 2;3 , 4;5A B C . Hy vit phng trnh cc ng thng cch u ba im A, B, C.

S: L cc ng trung bnh ABC

: 3 6 0

: 2 9 0

: 2 2 0

MN x y

NP x y

MP x y

.

+ Cao ng khi A, B nm 2005

Mt hnh thoi c: mt ng cho phng trnh l 2 7 0x y , mt cnh c phng

trnh l 3 3 0x y , mt nh l 0;1 . Tm phng trnh cc cnh ca hnh thoi.

+ Cao ng S Phm KomTum nm 2005

Trong mt phng vi h trc ta Oxy cho im 5

;22

M

v hai ng thng

1 : 2 0x y , 2 : 2 0x y . Lp phng trnh ng thng d qua M ct 1 2, ln lt ti A, B sao cho M l trung im ca on AB.

+ Cao ng S Phm Vnh Long khi A, B nm 2005

Trong mt phng vi h trc ta Oxy, cho ABC c 1;3A v hai ng trung tuyn xut pht t B v C ln lt c phng trnh: 2 1 0x y v 1 0y . Hy lp phng

trnh cc cnh ca ABC.

S: : 2 0, : 4 1 0, : 2 7 0AB x y BC x y CA x y .

+ Cao ng S Phm H Ni nm 2005

Trong mt phng vi h trc ta Oxy, cho ABC c im 1;2A , ng trung tuyn BM v ng phn gic trong CD tng ng c phng trnh 2 1 0x y , 1 0x y .

Hy vit phng trnh ng thng BC.

S: : 4 3 4 0BC x y .

+ Cao ng Bn Tre nm 2005

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Trong mt phng vi h trc ta Oxy, vit phng trnh cc cnh ca ABC bit nh

4; 1 ,A phng trnh mt ng cao v mt ng trung tuyn v cng mt nh ln lt l

1: 2 3 12 0d x y v

2: 2 3 0d x y .

S: :3 7 5 0, :3 2 10 0, :9 11 5 0AB x y AC x y BC x y .

+ Cao ng Kinh T K Thut Cn Th nm 2005

Trong mt phng vi h trc ta Oxy, cho ABC c nh 1;3 ,A phng trnh ng cao : 2 3 10 0BH x y v phng trnh ng thng :5 3 34 0BC x y . Xc nh

ta cc nh B v C.

S: 8;2 , 5; 3B C .

+ Cao ng S Phm H Nam khi H nm 2005

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho 1;2 , 5;4A B v

ng thng : 3 2 0x y . Tm im M trn ng thng sao cho MA MB

ngn

nht.

S: 5 3

;2 2

M

.

+ Cao ng S Phm Qung Ninh khi A nm 2005

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC c im

2; 1A v hai ng phn gic trong ca hai gc B, C ln lt c phng trnh : 2 1 0,B x y : 3 0C x y . Vit phng trnh cnh BC. S: : 4 3 0BC x y .

+ Cao ng S Phm in Bin khi A, B nm 2005

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC vung A. Bit ta 3;5 , 7;1A B v ng thng BC i qua im 2;0M . Tm ta nh C.

S: 3; 1C .

+ Cao ng S Phm C Mau khi A nm 2005

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho hai im

1;1 , 2;1A B v ng thng : 2 2 0d x y .

1/ Chng t rng hai im A, B v cng mt pha ca d.

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2/ Tm ta im M thuc d sao cho tng khong cch MA MB b nht.

S: 23 16

;15 13

M

.

+ Cao ng Truyn Hnh khi A nm 2005

Trong mt phng vi h trc ta Oxy, cho ABC c 0, 90AB AC BAC . Bit

1; 1M l trung im cnh BC v 2

;03

G

l trng tm ca ABC. Tm ta nh A,

B, C.

+ Cao ng Cng ng Vnh Long khi A, B nm 2005

Trong mt phng vi h trc ta Oxy, cho ABC c nh 3;0A v phng trnh hai ng cao ' : 2 2 9 0BB x y v ' : 3 12 1 0CC x y . Vit phng trnh cc cnh ca tam gic ABC.

+ Cao ng S Phm H Ni khi D1, T nm 2005

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC c 2; 4 ,A 0;2B v im C thuc ng thng: 3 1 0,x y din tch ABC bng 1 (n v din

tch). Hy tm ta im C.

S: 1 1

; 1; 22 2

C C

.

+ Cao ng Kinh T K Thut Cng Nghip I khi A nm 2006

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ba im 1;2 ,A 3;1 ,B 4;3C . Chng minh rng ABC l tam gic cn. Vit phng trnh cc ng

cao ca tam gic .

S: : 2 5 0, :3 10 0, : 2 5 0AH x y BI x y CK x y .

+ Cao ng Xy Dng s 2 khi A nm 2006

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho mt tam gic c mt nh l 4;3 ,A mt ng cao v mt ng trung tuyn i qua hai nh khc nhau c phng trnh ln lt l 3 11 0x y v 1 0x y . Hy vit phng trnh cc cnh

tam gic.

S: : 3 13 0, : 2 2 0, :7 29 0AC x y AB x y BC x y .

+ Cao ng Giao Thng Vn Ti III Tp. H Ch Minh khi A nm 2006

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Trong mt phng vi h trc ta Descarter vung gc Oxy, cho hnh thoi ABCD c phng trnh hai cnh v mt ng cho l : 7 11 83 0,AB x y

: 7 11 53 0,CD x y : 5 3 1 0BD x y . Tm ta B v D. Vit phng trnh ng

cho AC, ri suy ra ta ca A v C.

S: :3 5 13 0 4;5 , 6; 1AC x y A C .

+ Cao ng Bn Cng Hoa Sen khi A nm 2006

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho hai ng thng c phng trnh:

1 2: 2 3 1 0, : 4 5 0d x y d x y . Gi A l giao im ca d1 v d2. Tm

im B trn d1 v im C trn d2 sao cho ABC c trng tm l im 3;5G .

S: 61 43 5 55

1;1 , ; , ;7 7 7 7

A B C

.

+ Cao ng Kinh T K Thut Cn Th khi A nm 2006

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC vi 2;1 ,A 4; 3B v ; 2C m . nh m ABC vung ti C.

S: 1 5m m .

+ Cao ng in Lc Tp. H Ch Minh nm 2006

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ng thng d c

phng trnh 3 0x y v hai im 1;1 , 3;4A B . Tm ta im M thuc ng thng d sao cho khong cch t M n ng thng AB bng 1.

S: 0;3 10; 7M M .

+ Cao ng Kinh T Cng Ngh Tp. H Ch Minh khi D1 nm 2006

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC vung cn ti

4;1A v cnh huyn BC c phng trnh: 3 5 0x y . Vit phng trnh hai cnh gc vung AC v AB.

S: : 2 2 0AC x y v : 2 9 0AB x y .

+ Cao ng S Phm Bnh Phc nm 2006

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho 1;1 ,A 4;3B . Tm im C thuc ng thng 2 1 0x y sao cho khong cch t im C n ng

thng AB bng 6.

S: 43 27

; 7;311 11

C C

.

+ Cao ng S Phm Tr Vinh khi M nm 2006

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Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC bit 2; 4 ,C trong tm 0;4G v 2;0M l trung im cnh BC. Hy vit phng trnh ng thng cha cnh AB.

S: : 4 5 44 0AB x y .

+ Cao ng K Thut Cao Thng nm 2006

Trong mt phng vi h trc ta Oxy, cho ng thng :3 4 1 0d x y . Hy vit

phng trnh ng thng song song vi d v c khong cch n d bng 1.

S: 1 2: 3 4 4 0 :3 4 6 0x y x y .

+ Cao ng Kinh T Tp. H Ch Minh nm 2007

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho hai ng thng

1 2: 1 0, : 2 1 0d x y d x y v im 2; 4M . Vit phng trnh ng thng

i qua I v ct d1, d2 ln lt ti A v B m I l trung im ca AB.

S: : 4 14 0AB x y .

+ Cao ng Kinh T i Ngoi nm 2007

Trong mt phng ta Oxy, cho ABC. Bit im 4; 1 ,B ng cao AH c phng trnh l : 2 3 12 0,x y ng trung tuyn AM c phng trnh : 2 3 0x y . Vit

phng trnh cc ng thng cha cc cnh ca tam gic ABC.

S: 3;2 , 4;1 , 8; 7A B C .

+ Cao ng Xy Dng s 2 nm 2007

Vit phng trnh cc cnh ca ABC bit nh 1;1 ,A ng trung tuyn v ng cao i qua nh B ln lt c phng trnh: 3 4 27 0, 2 8 0x y x y .

S: : 1, : 2 1 0, : 8 49 0AB x AC x y BC x y .

+ Cao ng Cng Nghip Thc Phm nm 2007

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC c nh

2; 7 ,A trung tuyn CM, ng cao BK c phng trnh ln lt l 2 7 0x y v 3 11 0x y . Vit phng trnh cc ng thng AC v BC.

S: : 3 23 0AC x y v : 7 9 19 0BC x y .

+ Cao ng khi A, B, D nm 2008

Trong mt phng vi h trc ta Oxy, tm im A thuc trc honh v im B thuc trc tung sao cho A v B i xng vi nhau qua ng thng : 2 3 0d x y .

S: 2;0 , 0;4A B .

+ Cao ng A, B, D nm 2011 (Chng Trnh C Bn)

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Trong mt phng vi h to Oxy, cho ng thng : 3 0d x y . Vit phng trnh

ng thng i qua im 2; 4A v to vi ng thng d mt gc bng 045 .

S: 1 2: 4 0 : 2 0y x .

+ Cao ng A, B, D nm 2011 (Chng Trnh Nng Cao)

Trong mt phng vi h to Oxy, cho tam gic ABC c phng trnh cc cnh l : 3 7 0, : 4 5 7 0, :3 2 7 0AB x y BC x y CA x y . Vit phng trnh ng

cao k t nh A ca tam gic ABC.

S: :5 4 3 0AH x y .

+ i hc S PhmKinh tTi ChnhNng Nghip Tp. H Ch Minh nm 1977

Trong mt phng ta vung gc Oxy, hy vit phng trnh ng thng i qua giao im ca hai ng thng :3 5 2 0,x y 5 2 4 0x y v song song vi ng thng

2 4 0x y .

S: :38 19 30 0d x y .

+ i hc Th Dc Th Thao Tp. H Ch Minh nm 1977

Trong mt phng ta vung gc Oxy, hy lp phng trnh ng phn gic ca gc t to bi hai ng thng

1 2: 3 4 12 0, :12 3 7 0x y x y .

S: : 60 9 17 15 12 17 35 36 17 0d x y . + i hc Tng Hp Tp. H Ch Minh khi B nm 1978

Trong mt phng ta vung gc Oxy, cho hai ng thng d1 v d2 ln lt c phng trnh:

1 2: 1, : 3 3 0d x y d x y . Hy vit phng trnh ng thng d i xng vi

d2 qua ng thng d1.

S: :3 1 0d x y .

+ i hc Th Dc Th Thao Tp. H Ch Minh nm 1978

Trong mt phng ta vung gc Oxy, cho tam gic cn PRQ, bit phng trnh cnh y : 2 3 5 0,PQ x y cnh bn : 1 0PR x y . Tm phng trnh cnh bn RQ bit

rng n i qua im 1;1D . S: :17 7 24 0RQ x y .

+ i hc Bch Khoa i hc Tng Hp Tp. H Ch Minh nm 1979

Trong mt phng ta vung gc Oxy, cho ng cong 2: 9C y x v ng thng : 5 32 0d ax y .

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1/ V ng cong cho.

2/ Tnh khong cch z t mt im M ty ca ng cong n ng thng d theo

honh x ca M.

3/ Tnh khong cch ngn nht gia ng cong v ng thng.

S: 1/ V 2 2

2 2: 1

3 3

x yH trn Ox. 2/

24 5 9 32

41

x xz

. 3/

164;

5M

.

+ i hc Y Nha Dc Tp. H Ch Minh nm 1980

Trong mt phng ta vung gc Oxy, hy vit phng trnh ng thng i qua im

1;2A m khong cch t im 2;3M v im 4; 5N n ng thng y bng nhau.

S: :3 2 7 0 : 4 6 0d x y d x y .

+ Hc Vin Ngn Hng Tp. H Ch Minh nm 1991

Trong mt phng ta Descartes vung gc, cho ABC c nh 2;2A . Lp phng trnh cc cnh ca ABC. Bit rng cc ng thng 9 3 4 0x y v 2 0x y ln

lt l cc ng cao ca tam gic xut pht t B v C.

S: : 3 8 0, : 0, : 7 5 8 0AC x y AB x y BC x y .

+ i hc Cn Th 1993 i hc Hng Hi 1995 Trung Tm o To Cn B Y T Tp. H Ch Minh nm 1997 Hc Vin Hng Khng 2001

Lp phng trnh cc cnh ca tam gic ABC nu bit 2; 1 ,B ng cao qua A v ng phn gic trong gc C c phng trnh ln lt l 3 4 27 0; 2 5 0x y x y .

S: : 4 7 1 0, : 4 3 5 0, : 3AB x y BC x y AC y .

Li bnh

Phng trnh ng thng 2 5 0x y l phng trnh ng phn gic ngoi ca gc

C, khng phi l phng trnh ng phn gic trong gc C. ra thiu chnh xc. Mt

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s trng i hc ra ny tuyn sinh m khng pht hin ra, y, ti i li ng phn gic ngoi gc C l 2 5 0x y v gii ra kt qu nh trn.

+ Trung Tm o To Cn B Y T Tp. H Ch Minh nm 1993

Trong mt phng ta vung gc Oxy, cho hai im 2;5P v 5;1Q . Lp phng trnh ng thng qua P cch Q mt on c di bng 3 .

S: : 2 0 : 7 24 134 0d x d x y .

+ i hc Php L Tp. H Ch Minh nm 1994

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ba ng thng:

1 2 3: 3 4 6 0, : 4 3 1 0, : 0d x y d x y d y . Gi

1 2 2 3, ,A d d B d d

3 1C d d .

1/ Vit phng trnh ng phn gic trong ca gc A ca ABC v tnh din tch

ABC.

2/ Vit phng trnh ng trn ni tip ABC.

S: 1/ : 1 0A

d x y v 21

4

ABCS vdt

.

2/

2 2 2

73 9 1 73 73 1

8 8 8x y

.

+ i hc Tng Hp Tp. H Ch Minh khi A, B nm 1994

Trong mt phng ta vung gc Oxy, cho hai ng thng d1 v d2 ln lt c phng

trnh: 1: 0,d kx y k 2 22 : 1 2 1 0d k x ky k .

1/ Chng minh rng khi k thay i, ng thng d1 lun i qua mt im c nh.

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2/ Vi mi gi tr k, hy xc nh giao im ca d1 v d2.

3/ Tm qu tch ca giao im khi k thay i.

S: 1/ 1;0oM . 2/ 2 2

2 2

1 2;

1 1

k kM

k k

3/ ng trn: 2 2 1x y loi 1;0oM .

+ i hc Bch Khoa H Ni nm 1994

Phng trnh hai cnh mt tam gic trong mt phng ta l: 5 2 6 0;x y

4 7 21 0x y . Vit phng trnh cnh th ba ca tam gic, bit trc tm H trng vi

gc ta .

S: : 7 0BC y .

+ i hc M a Cht nm 1995

Lp phng trnh cc cnh ABC nu bit 1;3A v hai ng trung tuyn c phng trnh l 2 1 0x y v 1 0y .

S: : 2 7 0, : 2 0, : 4 1 0AB x y AC x y BC x y .

+ i hc Quc Gia H Ni nm 1995

Trn mt phng ta trc chun cho cc im 2;3 , 4; 1 , 3;5P Q R l cc trung im ca cc cnh ca mt tam gic. Hy lp phng trnh ca cc ng thng cha cc cnh ca tam gic .

S: : 6 7 3 0, : 2 1 0, : 2 5 3 0BC x y AB x y AC x y .

+ i hc S Phm H Ni 2 Khi A v i hc S Phm Quy Nhn nm 1995

Lp phng trnh cc cnh ca ABC trong mt phng vi h trc ta trc chun Oxy, nu cho 4; 5C v hai ng cao c phng trnh 5 3 4 0x y v 3 8 13 0x y . S: :3 5 13 0, :8 3 17 0, :5 2 1 0BC x y AC x y AB x y .

+ i hc Vn Ha H Ni nm 1995

Lp phng trnh cc cnh ca hnh vung bit rng hnh vung c nh l 4;8 v mt ng cho c phng trnh : 7 8 0x y .

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S: :3 4 32 0,AB x y : 4 3 1 0,AD x y : 4 3 24 0,BC x y

: 3 4 7 0CD x y .

+ i hc Y Khoa H Ni nm 1995

Trong mt phng ta vung gc Oxy, cho hai ng thng d1 v d2 ln lt c phng

trnh: 1 : 1d a b x y v 2 22 :d a b x ay b vi 2 2 0a b . Xc nh giao im ca d1 v d2, bin lun theo a, b s giao im y.

S: TH1. 1 2

1 2

0 // b a b d d

b a d d

. TH2. 1 2

10 0 ;

aa b d d A

b b

.

+ i hc Cn Th nm 1995

Trong mt phng ta vung gc Oxy, cho 2; 3 , 3; 2A B . Trng tm G ca ABC

nm trn ng thng :3 8 0,d x y din tch ABC bng 3

2. Tm ta im C.

S: 1; 1 2; 10C C .

+ i hc Ti Chnh H Ni nm 1996

Trong mt phng ta vung gc Oxy, cho ABC c 2;2M l trung im ca BC, cnh AB c phng trnh: 2 2 0,x y cnh AC c phng trnh: 2 5 3 0x y . Xc

nh ta cc nh ca ABC.

S: 4 7 40 11 76 25

; , ; , ;9 9 9 9 9 9

A B C

.

+ i hc Vn Lang t 1 khi B, D nm 1997

Trong mt phng ta Oxy, cho ng thng d c phng trnh 3 4 12 0x y .

1/ Xc nh ta qua cc giao im A, B ca d ln lt vi trc Ox, Oy.

2/ Tnh ta hnh chiu H ca gc O trn ng thng d.

3/ Vit phng trnh ng thng d' i xng vi d qua O.

www.VNMATH.com

S: 1/ 4;0 , 0;3A B . 2/ 36 48

;25 25

H

. 3/ ' : 3 4 12 0d x y .

+ i hc An Ninh 2 khi D nm 1997

Trong mt phng ta cho im 0;2A v im ; 2B m . Hy vit phng trnh ng thng trung trc d ca AB. Chng minh rng d lun tip xc vi ng cong C c nh khi m thay i.

S: 2

: ,4 8

m md y x lun tip xc vi parabol 2

1:

8P y x .

+ i hc Hu khi D nm 1997

Trong mt phng ta Oxy, cho hai ng thng 1: 4 3 12 0,x y

2: 4 3 12 0x y .

1/ Tm ta cc nh ca tam gic c ba cnh ln lt nm trn cc ng thng 1, 2

v trc tung.

2/ Xc nh tm v bn knh ng trn ni tip ca tam gic ni trn.

S: 1/ 0; 4 , 0;4 , 3;0A B C . 2/ 4 4

;0 , 3 3

E r

.

+ i hc S Phm H Ni 2 nm 1997

Trong mt phng ta Oxy, cho 2;1 , 0;1 , 3;5 , 3; 1A B C D .

1/ Tnh din tch t gic ABCD.

2/ Vit phng trnh cc cnh ca hnh vung c hai cnh song song i qua A v C v hai cnh cn li i qua B v D.

www.VNMATH.com

S: 1/ 6 ABCDS vdt . 2/

: 7 15 0 : 3 1 0

: 7 26 0 : 3 12 0

: 7 7 0 :3 1 0

: 7 4 0 :3 10 0

MN x y MN x y

PQ x y PQ x y

NP x y NP x y

MQ x y MQ x y

.

+ i hc Y Dc Tp. H Ch Minh h C nhn nm 1997

Cho ABC, cnh BC c trung im 0;4 ,M cn hai cnh kia c phng trnh l 2 11 0x y v 4 2 0x y .

1/ Xc nh nh A.

2/ Gi C l nh nm trn ng thng 4 2 0x y v N l trung im AC. Tm ta

im N ri tnh ta B, C.

S: 1/ 6; 1A . 2/ 2;1 , 2;7C B .

+ i hc Quc Gia Tp. H Ch Minh khi A i hc Lut nm 1997

Trong mt phng ta vung gc Oxy, cho ng thng : 2 4 0d x y v hai im

3;3 , 5; 19M N . H MK d v gi P l im i xng ca M qua d.

1/ Tm ta im K v P.

2/ Tm im A trn d sao cho AM AN c gi tr nh nht v tnh gi tr .

www.VNMATH.com

S: 1/ 1;2 , 1;1K P . 2/ min

3;10 2 85A AM AN .

+ i hc Lt nm 1998

Trong mt phng ta vung gc Oxy, cho hai ng thng 1 : 1 2 1 0d a x y a v 22 : 1 0d x a y a .

1/ Tm giao im I ca d1 v d2.

2/ Tm a ng thng qua 0; , ;0M a N a cng i qua im I.

S: 1/

22

2 2

1 13 1;

1 1

a aaI

a a

. 2/ 1

1 2

a a .

+ i hc K Thut Cng Ngh Tp. H Ch Minh khi B, D nm 1998

Trong mt phng ta Oxy, cho ABC c 2;2A , bit tam gic c hai ng cao l: 9 3 4 0x y v 2 0x y .

1/ Vit phng trnh cc ng trung trc ca tam gic ABC.

2/ Vit phng trnh ng trn ngoi tip ABC v xc nh ta trng tm ABC.

S: 1/

: 3 3 8 0

:9 3 3 0

:15 21 41 0

AB

AC

BC

d x y

d x y

d x y

. 2/ 2 2

5 9 185:

12 4 72C x y

v

5 17;

9 9G

.

www.VNMATH.com

+ i hc Giao Thng Vn Ti Tp. H Ch Minh 1 nm 1998

Trong mt phng ta Oxy, cho hai im 1;2A v 3;4B . Tm ta im C trn ng thng: 2 1 0x y sao cho ABC vung C.

S: 3 4

3;2 ;5 5

C C

.

+ i hc Nng khi A nm 1998

Trong mt phng ta vung gc Oxy, cho im 3;0P v hai ng thng:

1: 2 2 0d x y v

2: 3 0d x y . Gi d l ng thng qua P v ct

1 2,d d ln lt

A v B. Vit phng trnh ca d bit rng PA PB .

S: : 4 5 12 0 :8 24 0d x y d x y .

+ i hc Vn Lang khi B, D nm 1998

Trong mt phng ta vung gc Oxy, cho ABC c nh 3;5 ,B ng cao k t A c phng trnh: 2 5 3 0x y v ng trung tuyn k t nh C c phng trnh:

5 0x y . Tm ta nh A v vit phng trnh cc cnh ca tam gic.

S: 1;1 , :5 2 25 0, : 2 1 0, : 4 5 0A BC x y AB x y AC x y .

+ i hc Quc Gia Tp. H Ch Minh t 3 nm 1998

Trong mt phng ta vung gc Oxy, cho ABC c trng tm 2; 1G v cc cnh : 4 15 0AB x y v : 2 5 3 0AC x y .

1/ Tm ta nh A v ta trung im M ca BC.

2/ Tm ta nh B v vit phng trnh ng thng BC.

S: 1/ 4;1 , 1;2A M . 2/ 3; 3 , : 2 3 0B BC x y .

+ i hc Hng Hi nm 1998

Trong mt phng ta vung gc Oxy, cho 1;1 , 1;3A B v ng thng d c phng trnh : 4 0d x y .

www.VNMATH.com

1/ Tm trn d im C cch u hai im A, B.

2/ Vi C va tm c, tm D sao cho ABCD l hnh bnh hnh. Tnh din tch hnh bnh

hnh.

S: 1/ 3; 1C . 2/ 1; 3D v 12 ABCDS vdt .

+ i hc Cn Th nm 1998

Trong mt phng ta vung gc Oxy, cho ABC c nh 1; 3A .

1/ Bit ng cao :5 3 25 0,BH x y ng cao :3 8 12 0CK x y . Tm ta

nh B, C.

2/ Bit ng trung trc ca AB l :3 2 4 0x y v trong tm 4; 2G . Tm ta

nh B, C.

S: 1/ 2;5 , 4;0B C . 2/ 5;1 , 8; 4B C .

+ i hc Vn Ha H Ni nm 1998

Trong mt phng ta vung gc Oxy, cho ABC bit nh 4; 1C v ng cao, ng trung tuyn k t mt nh c phng trnh ln lt l 2 3 12 0x x v 2 3 0x y . Tm phng trnh cc cnh ca tam gic ABC.

S: :3 7 5 0, :3 2 10 0, :9 11 5 0AC x y BC x y AB x y .

+ i hc Hu khi D nm 1998

www.VNMATH.com

Trong mt phng ta vung gc Oxy, hy vit phng trnh ng thng song song vi :3 4 1 0d x y v c khong cch n ng ng thng d bng 1.

S: :3 4 4 0 :3 4 6 0x y x y .

+ i hc Kin Trc H Ni nm 1998

Trong mt phng ta vung gc Oxy, cho ba im 2;4 , 3;1 , 1;4A B C v ng thng d c phng trnh: : 1 0d x y .

1/ Tm M d sao cho AM MB nh nht.

2/ Tm N d sao cho AN CN nh nht.

S: 1/ min

11 7; 10

4 4M AM BM

. 2/

min

23 16; 5

7 7N AN CN

.

+ i hc Dn Lp K Thut Cng Ngh khi D nm 1999

Trong mt phng ta vung gc Oxy, cho im 2;3M . Tm phng trnh ng thng d qua M v cch u hai im 1;0 , 2;1A B . S: : 3 11 0 : 1 0d x y d x y .

+ i hc Cn Th khi A nm 1999

Trong mt phng ta vung gc Oxy, cho ba im 3;4 , 5; 1 , 4;3A B C .

1/ Tnh di , , AB BC AC . Hy cho bit tnh cht (nhn, t, vung) ca cc gc trong

ABC.

2/ Tnh di ng cao AH ca ABC v vit phng trnh ng thng AH.

www.VNMATH.com

S: 1/ 29, 50, 97AB AC BC nhn.

2/ 37

, :9 4 11 097

AH AH x y .

+ i hc M Thut Cng Nghip khi A nm 1999

Trong mt phng ta vung gc Oxy, cho hai ng thng c phng trnh ln lt l:

1: 1 0,d x y

2: 3 1 0d x y v im 1;2M . Vit phng trnh ng thng d i

qua im M, ct d1 v d2 ln lt ti M1, M2 v tha mt trong cc iu kin sau:

1/ 1 2

MM MM .

2/ 1 2

2MM MM .

S: 1/ 2: 1 0d MM x . 2/

2: 3 0d MM x y .

+ i hc Dc H Ni nm 1999

Trong mt phng ta vung gc Oxy, cho hai ng thng 1 : 1d a b x y v

2 22 :d a b x ay b vi 2 24 1b a .

1/ Xc nh giao im ca d1 v d2.

www.VNMATH.com

2/ Tm tp hp E cc giao im ca d1 v d2 khi a, b thay i.

S: 1/ 1

;a

Mb b

. 2/ 2 2

22 : 1

1 1

2

x yEllipse E

.

+ i hc Nng khi A i hc Kinh T Tp. H Ch Minh nm 1999

Trong mt phng ta vung gc Oxy, cho hai ng thng 1: 2 2 0d x y v

2: 2 4 7 0d x y .

1/ Vit phng trnh ng phn gic ca gc to bi d1 v d2.

2/ Vit phng trnh ng thng qua im 3;1P cng vi d1, d2 to thnh tam gic cn

c nh l giao ca d1 v d2.

S: 1/ : 2 6 3 0

: 6 2 11 0

d x y

d x y

. 2/ : 3 10 0

: 3 0

x y

x y

.

+ i hc S Phm H Ni 2 nm 1999

Trong mt phng ta vung gc Oxy, cho tam gic ABC vi cc nh 6; 3 ,A 4;3B v 9;2C .

1/ Vit phng trnh ng thng d cha ng phn gic trong ca gc A ca ABC.

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2/ Tm im P trn ng thng d sao cho t gic ABCP l hnh thang.

S: 1/ : 3 0d x y . 2/ 14;17 2;5P P .

+ i hc Kinh T Quc Dn H Ni nm 1999

Trong mt phng ta vung gc Oxy, hy vit phng trnh ng thng i qua im

0;1A v to vi ng thng 2 3 0x y mt gc bng 045 . S: :3 1 0 : 3 1 0d x y d x y .


Cho ABC c 2; 1A v phng trnh cc ng cao l 2 1 0, 3 2 0x y x y . Lp phng trnh ng trung tuyn ca tam gic qua nh A.

S: : 32 30 0AM x y .

+ i hc M Bn Cng Tp. H Ch Minh khi A, B nm 2000

Trong mt phng ta vung gc Oxy, cho im 1;6M v : 2 3 3 0d x y .

1/ Vit phng trnh ng thng d1 i qua M v song song vi d.

2/ Vit phng trnh ng thng d2 i qua M, vung gc vi d v xc nh ta hnh

chiu vung gc ca M ln ng thng d.

S: 1/ 1: 2 3 16 0d x y . 2/

2: 3 2 15 0d x y v 3;3H .

+ i hc Ty Nguyn khi D nm 2000

Trong mt phng ta vung gc Oxy, hy lp phng trnh tng qut ca ng thng i qua im 2;3I v cch u hai im 5; 1A v 3;7B .

www.VNMATH.com

S: : 4 5 0 : 3 0d x y d y .

+ i hc S Phm H Ni 2 khi A nm 2000

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC v nh 1;1A . Cc ng cao h t B v C ln lt nm trn cc ng thng d1 v d2 theo th t c phng trnh 2 8 0x y v 2 3 6 0x y . Hy vit phng trnh ng thng cha

ng cao h t nh A v xc nh ta nh B, C ca ABC.

S: 1 29 7

; , :10 13 23 0, 17; 26 , 3;04 2

I d d AI x y B C

.

+ i hc Thng Mi nm 2000

Trong mt phng ta vung gc Oxy, cho ABC c 2; 1A v phng trnh hai ng phn gic trong ca gc B v gc C ln lt c phng trnh: : 2 1 0

Bd x y v

: 3 0C

d x y . Tm phng trnh ca ng thng cha cnh BC.

S: : 4 3 0BC x y .

+ Hc Vin Ngn Hng Tp. H Ch Minh nm 2000

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho ABC c nh

2; 4C v trng tm 0;4G .

1/ Gi s 2;0M l trung im ca cnh BC. Xc nh ta cc nh A v B.

2/ Gi s M di ng trn ng thng : 2 0,D x y tm qu tch im B. Hy xc

nh M di cnh AB l ngn nht.

S: 1/

4;123

6;42

AMA MG

BCB CM

. 2/ Qu tch l : 10 0d x y v 1 9

;4 4

M

.

+ i hc Giao Thng Vn Ti khi A nm 2001

Trong mt phng vi h trc ta Descarter vung gc Oxy, cho hnh bnh hnh ABCD

c s o din tch bng 4. Bit ta cc nh 1;0 , 2;0A B v giao im I ca hai ng cho AC v BD nm trn ng thng y x . Hy tm ta cc nh C v D.

S: 3;4 , 2;4 5; 4 , 6; 4C D C D .

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+ Hc Vin K Thut Qun S nm 2001

Trong mt phng vi h trc ta Oxy, cho ABC cn, cnh y BC c phng trnh 3 1 0x y . Cnh bn AB c phng trnh 5 0x y . ng thng cha cnh AC i

qua im 4;1M . Tm ta nh C. S: Ba cnh ABC ng quy ti M V l Bi ton khng xc nh C tha

yu cu bi ton.

+ i hc Nng Nghip I nm 2001

Trong mt phng Oxy cho im 1;1A v ng thng d c phng trnh: 4 3 12x y .

1/ Gi B v C ln lt l giao im ca d vi cc trc Ox v Oy. Xc nh ta trc

tm ca ABC.

2/ im M chy trn ng thng d. Trn na ng thng i qua hai im A v M, ly

im N sao cho . 4AM AN

. im N chy trn ng cong no ? Vit phng trnh ng cong .

S: 1/ 3; 2H . 2/ N chy trn ng trn 2 2

13 11: 4

5 5C x y

.


Trong mt phng vi h trc ta Oxy, cho im 5

;22

M

v hai ng thng c

phng trnh ; 2 02

xy y x . Lp phng trnh ng thng d qua M v ct hai ng

thng ni trn ti hai im A v B sao cho M l trung im ca AB.

S: 2y .

+ i hc Hu khi A,B,V nm 2001

Vit phng trnh ba cnh ca ABC trong mt phng ta Oxy, cho bit nh 4;3 ,C ng phn gic trong v ng trung tuyn k t mt nh ca tam gic c phng trnh ln lt l 2 5 0x y v 4 13 10 0x y .

S: : 7 0, : 7 5 0, : 8 20 0AC x y AB x y BC x y .

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+ i hc Kinh T Quc Dn nm 2001

Trong mt phng vi h trc ta trc chun Oxy, hy lp phng trnh cc cnh ca ABC nu cho 4;5B v hai ng cao h t hai nh cn li ca tam gic ln lt c phng trnh: 5 3 4 0x y v 3 8 13 0x y .

S: :3 5 37 0, :8 3 47 0, : 2535 3016 29033 0BC x y AB x y AC x y .

+ i hc khi A nm 2002

Trong mt phng vi h ta cc vung gc yOx , xt tam gic ABC vung ti A,

phng trnh ng thng BC l 3 3 0x y , cc nh A v B thuc trc honh v bn

knh ng trn ni tip bng 2. Tm ta trng tm G ca tam gic ABC.

S: 1 2

7 4 3 6 3 4 3 1 6 2 3; , ;

3 3 3 3G G

.

+ i hc khi B nm 2002

Trong mt phng vi h ta cc vung gc yOx , cho hnh ch nht ABCD c tm

1; 0

2I

, phng trnh ng thng AB l 2 2 0x y v 2 DAB A . Tm ta cc

nh A, B, C, D bit rng nh A c honh m.

S: 2;0 , 2;2 , 3;0 , 1; 2A B C D .


Trong mt phng vi h ta cc vung gc yOx , cho tam gic ABC c

,AB AC 090BAC . Bit 1; 1M l trung im cnh BC v 2

; 03

G

l trng tm

tam gic ABC. Tm ta cc nh A, B, C.

S: 0;2 , 4;0 , 2 2A B C .

+ D b 1 i hc khi D nm 2003

Trong mt phng vi h ta yOx , cho tam gic ABC c nh 1;0A v hai ng thng ln lt cha cc ng cao v t B v C c phng trnh tng ng l:

2 1 0,x y 3 1 0x y . Tnh din tch tam gic ABC.

S: 5; 2 , 1;4 14ABCB C S .


Trong mt phng vi h ta yOx , cho hai im 0;2A v 3; 1B . Tm ta trc tm v ta tm ng trn ngoi tip ca tam gic OAB.

S: 3; 1 , 3;1H I .

www.VNMATH.com


Trong mt phng vi h ta yOx , cho hai im 1;1 , 4; 3A B . Tm im C thuc ng thng 2 1 0x y sao cho khong cch t C n ng thng AB bng 6.

S: 1 243 27

7;3 , ;11 11

C C

.

+ i hc khi D nm 2004

Trong mt phng vi h ta yOx , cho ABC c cc nh 1;0 , 4;0 , 0;A B C m vi 0m . Tm ta trng tm G ca ABC theo m. Xc nh m tam gic GAB vung ti

G.

S: 1; , 3 63

mG m

.

+ D b 2 i hc khi A nm 2004

Trong mt phng vi h ta yOx , cho im 0;2A v ng thng : 2 2 0d x y . Tm trn d hai im B, C sao cho tam gic ABC vung B v 2AB BC .

S: 2 6

;5 5

B

.

+ D b 1 i hc khi B nm 2004

Trong mt phng vi h ta yOx , cho im 2;0I v hai ng thng

1: 2 5 0,d x y

2: 3 0d x y . Vit phng trnh ng thng d i qua im I v ct

hai ng thng d1, d2 ln lt ti A, B sao cho 2IA IB

.

S: : 7 3 6 0d x y .


Trong mt phng vi h ta yOx , cho tam gic ABC vung A. Bit

1;4 , 4; 1 ,A B ng thng BC i qua im 7

;23

K

. Tm to nh C.

S: 69 1579

;50 250

C

.


Trong mt phng vi h ta Oxy, cho im 2;3A v hai ng thng

1: 5 0,d x y

2: 2 7 0d x y . Tm to cc im B trn d1 v C trn d2 sao cho tam

gic ABC c trng tm 2;0G .


www.VNMATH.com

Trong mt phng vi h ta Oxy, cho hai ng thng 1: 0d x y v

2: 2 1 0d x y .

Tm to cc nh hnh vung ABCD bit rng nh A thuc d1, nh C thuc d2 v cc nh B, D thuc trc honh.

S: 1;1 , 0,0 , 1; 1 , 2;0A B C D hoc 1;1 , 2,0 , 1; 1 , 0;0A B C D .


Trong mt phng vi h ta Oxy, cho tam gic ABC cn ti nh A c trng tm

4 1;

3 3G

, phng trnh ng thng BC l 2 4 0x y v phng trnh ng thng

BG l 7 4 8 0x y .Tm ta cc nh A, B, C.

S: 0;3 , 0;2 , 4;0A B C .


Trong mt phng vi h ta Oxy, cho cc ng thng ln lt c phng trnh:

1: 3 0,d x y

2: 4 0,d x y

3: 2 0d x y . Tm to im M nm trn ng

thng d3 sao cho khong cch t M n ng thng d1 bng hai ln khong cch t M n ng thng d2.

S: 22; 11 2;1M M .


Trong mt phng vi h ta Oxy, cho tam gic ABC c nh A thuc ng thng : 4 2 0d x y , cnh BC song song vi d. Phng trnh ng cao : 3 0BH x y

v trung im ca cnh AC l 1;1M . Tm to cc nh A, B, C.

S: 2 2 8 8

; , 4;1 , ;3 3 3 3

A B C

.


Trong mt phng vi h ta Oxy, cho tam gic ABC cn ti B, vi 1; 1 , 3;5A B . im B nm trn ng thng : 2 0d x y . Vit phng trnh cc ng thng AB, BC.

S: : 23 24 0AB x y v :19 13 8 0BC x y .


Trong mt phng vi h ta Oxy, cho tam gic ABC c nh 2;1A , ng cao qua nh B c phng trnh 3 7 0x y v ng trung tuyn qua nh C c phng trnh

1 0x y . Xc nh to cc nh B v C ca tam gic.

S: 2; 3 , 4; 5B C .


www.VNMATH.com

Trong mt phng vi h ta Oxy, cho im 2;2A v cc ng thng:

1: 2 0,d x y

2: 8 0d x y . Tm to cc im B v C ln lt thuc d1 v d2 sao

cho tam gic ABC vung cn ti A.

S: 1;3 , 3;5 3; 1 , 5;3B C B C .


Trong mt phng vi h ta Oxy, cho tam gic ABC c trng tm 2;0 ,G phng trnh cc cnh : 4 14 0, : 2 5 2 0AB x y AC x y . Tm to cc nh A, B, C.

S: 4;2 , 3; 2 , 1;0A B C .


Trong mt phng vi h ta Oxy, cho im 2;1A . Trn trc Ox, ly im B c honh 0

Bx , trn trc Oy, ly im C c tung 0

Cy sao cho tam gic ABC vung ti A.

Tm cc im B, C sao cho din tch tam gic ABC ln nht.

S: 0;0 , 0;5B C .


Trong mt phng vi h ta Oxy, cho cc im 0;1 , 2; 1A B v cc ng thng 1 : 1 2 2 0,d 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 l giao im ca d1 v d2. Tm m sao cho PA + PB ln nht.

S: Ch : 2 2 2 22 2A 16PA PB PA PB B . Do max

4PA PB khi P

l trung im ca cung AB. Khi 2;1P hay 0; 1 1 2P m m .


Trong mt phng vi h ta Oxy, hy xc nh to nh C ca tam gic ABC bit rng hnh chiu vung gc ca C trn ng thng AB l im 1; 1 ,H ng phn gic trong gc A c phng trnh 2 0x y v ng cao k t B c phng trnh

4 3 1 0x y .

S: 10 3

;3 4

C

.

+ i hc khi A nm 2009 (Chng trnh c bn)

Trong mt phng vi h ta Oxy, cho hnh ch nht ABCD c im 6;2I l giao im ca hai ng cho AC v BD. im 1;5M thuc ng thng AB v trung im E ca cnh CD thuc ng thng : 5 0x y . Vit phng trnh ng thng AB.

S: 5 0

:4 19 0

yAB

x y

.

www.VNMATH.com

+ i hc khi B nm 2009 (Chng trnh nng cao)

Trong mt phng vi h ta Oxy, cho tam gic ABC cn ti A c nh 1;4A v cc nh B, C thuc ng thng : 4 0x y . Xc nh to cc im B v C, bit din

tch tam gic ABC bng 18.

S: 11 3 3 5 3 5 11 3

; , ; ; , ;2 2 2 2 2 2 2 2

B C B C

.

+ i hc khi D nm 2009 (Chng trnh c bn)

Trong mt phng vi h ta Oxy, cho tam gic ABC c 2;0M l trung im ca cnh AB. ng trung tuyn v ng cao qua nh A ln lt c phng trnh l 7 2 3 0,x y 6 4 0x y . Vit phng trnh ng thng AC.

S: :3 4 5 0AC x y .

+ i hc khi A nm 2010 (Chng trnh nng cao)

Trong mt phng ta Oxy, cho tam gic ABC cn ti A c nh 6;6 ;A ng thng i qua trung im ca cc cnh AB v AC c phng trnh 4 0x y . Tm to cc

nh B v C, bit im 1; 3E nm trn ng cao i qua nh C ca tam gic cho.

S: 0; 4 , 4;0 6;2 , 2; 6B C B C .

+ i hc khi B nm 2010 (Chng trnh c bn)

Trong mt phng ta Oxy, cho tam gic ABC vung ti A, c nh 4;1 ,C phn gic trong gc A c phng trnh 5 0x y . Vit phng trnh ng thng BC, bit din

tch tam gic ABC bng 24 v nh A c honh dng.

S: :3 4 16 0BC x y .

+ i hc khi D nm 2010 (Chng trnh nng cao)

Trong mt phng ta Oxy, cho im 0;2A v l ng thng i qua O. Gi H l hnh chiu vung gc ca A trn . Vit phng trnh ng thng , bit khong cch t H n trc honh bng AH.

S: 12 : 5 1 2 5 2. 0x y . + i hc khi B nm 2011 (Chng trnh c bn)

Trong mt phng to Oxy, cho hai ng thng : 4 0x y v : 2 2 0d x y .

Tm ta im N thuc ng thng d sao cho ng thng ON ct ng thng ti im M tha mn . 8OM ON .

S: 6 2

0; 2 ;5 5

N N

.

+ i hc khi D nm 2011 (Chng trnh c bn)

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Trong mt phng ta Oxy, cho tam gic ABC c nh 4;1 ,B trng tm 1;1G v ng thng cha phn gic trong ca gc A c phng trnh 1 0x y . Tm ta cc

nh A v C.

S: 4;3 , 3; 1A C .

+ i hc khi A nm 2012 (Chng trnh c bn)

Trong mt phng vi h ta Oxy, cho hnh vung ABCD. Gi M l trung im ca

cnh BC, N l im trn cnh CD sao cho 2CN ND . Gi s 11 1

;2 2

M

v ng thng

AN c phng trnh 2 3 0x y . Tm ta im A.

S: 1; 1 4;5A A .

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Documents

Cac Bai Toan Duong Thang Oxy 1997-2012