[DeThiThu.Net]50bai-oxy-loi-giai-chi-tiet.pdf

8/17/2019 [DeThiThu.Net]50bai-oxy-loi-giai-chi-tiet.pdf

1/42

CHUYÊN Đ LUYN THI ĐI HC MÔN TOÁN

www.facebook.com/dethithu.net

TUYN TP HÌNH HC GII TÍCHTRONG MT PHNG HAY VÀ ĐCSC

(phiên bn 1)

Giáo viên : Nguyn Minh Tin

1

http://dethithu.net

*Truy cập http://dethithu.net mỗi ngày để tải các đề thi thử THPT Quốc Gia ( Đại Học )các môn TOÁN – ANH – VĂN – LÝ – HÓA – SINH mới nhất,nhanh nhất từ các trườngTHPT và trung tâm luyện thi đại học trong nước.Chúng tôi luôn cập nhật đề thi thử mỗi

ngày vậy nên các bạn yên,luôn có các đề thi thử mới nhất để các bạn tham khảo.

*Tham gia nhóm : ÔN THI ĐH TOÁN – ANH trn Facebook đ cng hi đáp, hctp : http://facebook.com/groups/onthidhtoananhvan*Like Fanpage : Đ Thi Th THPT Quc Gia – Ti Liu Ôn Thi đ cp nht nhiuhn qua Facebook. http://facebook.com/dethithu.net

http://www.facebook.com/dethithu.nethttp://dethithu.net/http://dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.facebook.com/dethithu.net


2/42


3/42

w w w

. d e t

i t u . n

e t

HÌNH HC GII TÍCH TRONG MT PHNG

Đt I A = I B = x ⇒ S IAB = 12

x2; S IAD = x√

5 = S IBC ; S ICD = 10

⇒ S ABCD = 12

x2 + 2x√

5 + 10 = 45

2 ⇔

x =

√ (5 tm)

x = −5√ (5 loai)

⇒ DI IB

= 2 ⇒ −→DI = 2−→IB (∗)

Gi B (b; 2b − 1) ∈ BD t (∗) ⇒ B (3;5)

Phương trình đưng thng BC đi qua B và C ⇒ BC : 4x + 3y − 27 = 0.

Bài toán gii quyt xong.

Đ bài 03 (k2pi Ln 15 - 2014) : Trong h ta đ Oxy cho hình vuông ABCD có phương trìnhđưng thng AD là (d) : 3x

−4y

−7 = 0. Gi E là đim nm bên trong hình vuông ABCD sao cho

tam giác EBC cân có BE C = 150o. Vit phương trình đưng thng AB bit đim E (2; −4).

Li gii tham kho :

Tam giác BEC cân và có BE C = 150o ⇒ tam giác BEC cân ti E

Gi H là hình chiu ca E lên AD ⇒ H là trung đim ca AD và H E = d (E ; AD) = 3

Đt cnh hình vuông là AB = x

Tam giác BEC cân ti E có BEC = 150o ⇒ EB C = 15o. Gi I là trung đim ca BC ⇒ BI = x2

; EI =

x − 3Tam giác BIE vuông ti I có góc EB I = 15o ⇒ tan 15o = EI

BI =

2x − 6x

Tng hp các bài toán đc sc 3

http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


4/42


5/42


6/42

w w w

. d e t

i t u . n

e t


A là giao đim ca (d1) và (d2) ⇒ ta đ đim A (−2; −1)Qua M k đưng thng ⊥(d2) ct (d2) ti I và AC ti NMN qua M và ⊥(d2) ⇒ (M N ) : x + y − 1 = 0I là giao đim ca MN và (d2) ⇒ I (0; 1)I là trung đim ca MN ⇒ N (−1;2)

Phương trình đưng thng (AB) : x − 3y − 1 = 0 và (AC ) : 3x − y + 5 = 0

Đim B ∈ AB ⇒ B (3a + 1; a), đim C ∈ AC ⇒ C (b; 3b + 5)

Ta có B C ⊥AH ⇔ −−−→AH ⊥−−−→BC ⇔ −−−→AH.−−−→BC = 0−−−→

AH = (2; 1) ; −−−→BC = (b

−3a

−1; 3b + 5

−a)

⇒ 2 (b − 3a − 1) + (3b + 5 − a) = 0 ⇔ 5b − 7a (1)+ 3 = 0

Ta có S ABC = 1

2d (C,AB) .AB =

|8b + 14|√ 10

.

(3a + 3)2 + (a + 1)2 = 180

7 (2)

T (1) và (2) ⇒

a =

8

7

a = −227

thay ngưc li ta có các đim A, B, C.


Đ bài 07 : Trong h trc ta đ Oxy cho tam giác ABC vuông ti A có AC = 2AB, phương

trình đưng thng cha cnh AB có phương trình là (d) : 2x − y + 7 = 0, đim G

0; 1

3

là trng

tâm ca tam giác ABC. Tìm ta đ các đnh ca tam giác ABC bit đnh B có hoành đ bé hơn

−2.

Li gii tham kho :

Gi M là trung đim ca AC ⇒ AM = M C = AB ⇒ ∆BAM vuông cân ti A ⇒ M BA = 45o


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


7/42

w w w

. d e t

i t u . n

e t


Gi −→n1 là vtpt ca đưng thng (d) ⇒ −→n1 = (2; −1) và −→n2 = (a; b) là vtpt ca đưng thng BG⇒ cos(−→n1, −→n2) =

√ 2

2 ⇒ |2a − b|√

5.√

a2 + b2=

√ 2

2

⇔ 3a2 − 8ab − 3b2 = 0 ⇔ a = 3b

a = −13

b

Vi a = 3b chn −→n2 = (3;1) ⇒ đưng thng BG qua G có vtpt −→n2 ⇒ BG : 9x + y − 1 = 0

B là giao đim ca AB và BG ⇒

x = −43

y = 13

3

loi do hoành đ đim B nh hơn −2

Vi a = −b

3 chn −→n2 = (1; −3) ⇒ đưng thng BG qua G có vtpt −→n2 ⇒ BG : x − 3y + 1 = 0B là giao đim ca AB và BG ⇒ B (−4; −1) ( tha mãn )

M là trung đim ca AC ⇒ M (3a − 1; a) ∈ BG ta có −−−→BG = 23

−−→BM ⇒ M (2; 1)

Phương trình đưng thng AC đi qua đim M và vuông góc vi AB ⇒ AC : x + 2y − 4 = 0

Ta đ đim A là giao đim AC và AB ⇒ A (−2;3) ⇒ C (6; −1)


Đ bài 08 ( k2pi Ln 14 - 2014) : Trong h trc ta đ Oxy cho tam giác ABC có đim

B

1

2; 1

. Đưng tròn ni tip tam giác ABC tip xúc vi các cnh BC, CA và AB ti D, E và F.

Bit đim D (3;1) và phương trình đưng thng EF có phương trình là (d) : y − 3 = 0. Tìm ta đđnh A bit đnh A có tung đ không âm.

Li gii tham kho :

Phương trình đưng thng BC đi qua đim B và D ⇒ BC : y − 1 = 0 ⇒ BC//EF

Do đó tam giác ABC cân ti A và D chính là trung đim ca BC.Phương trình đưng thng AD đi qua D và vuông góc vi BC ⇒ AD : x − 3 = 0


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


8/42

w w w

. d e t

i t u . n

e t


Đim E (a; 3) ∈ EF ta có B E = BD ⇒

a − 12

2

+ 22 = 25

4 ⇔

a − 1

2

2

= 9

4 ⇔

a = 2

a = −1

a = 2

⇒ phương trình AB đi qua đim B và E

⇒AB : 4x

−3y + 1 = 0

A là giao đim ca AB và AD ⇒ A

3; 13

3

a = −1 ⇒ phương trình AB đi qua đim B và E ⇒ AB : 4x + 3y − 5 = 0

A là giao đim ca AB và AD ⇒ A

3; −73

( loi)

Vy đim A

3; 13

3

Đ bài 09 : Trong h trc ta đ Oxy cho hình ch nht ABCD có AD = 2AB đim A (1;5),phư ơng trình đưng chéo BD là 3x + 4y − 13 = 0. Tìm ta đ các đnh còn li ca hình ch nhtbit B có hoành đ âm.

Li gii tham kho :

Xét tam giác ABD vuông ti A có B D2 = AB2 + AD2 = 5AB2 ⇒ BD = AB√ 5

⇒ cos ABD = ABBD

= 1√

5

Phương trình đưng chéo BD có vtpt −→n1 = (3;4). Gi

−→n = (a; b) là vtpt ca đưng thng AB

⇒ cos ABD = |3a + 4b|5.

√ a2 + b2

= 1√

5⇔ 4a2 + 24ab + 11b2 = 0 ⇔

a = −

11

2 b

a = −12

b

Vi a = −112

b chn −→n = (11; −2) ⇒ đưng thng AB có phương trình 11x − 2y − 1 = 0

Ta đ đim B là giao đim ca AB và BD ⇒ B

3

5; 14

5

loi do B có hoành đ âm.

Vi a = −12

b chn −→n = (1; −2) ⇒ đưng thng AB có phương trình x − 2y + 9 = 0

Ta đ đim B là giao đim ca AB và BD ⇒ B (−1;4) ( tha mãn )


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


9/42

w w w

. d e t

i t u . n

e t


Phương trình đưng thng AD đi qua đim A và vuông góc vi AB ⇒ AD : 2x + y − 7 = 0

Ta đ đim D là giao đim ca AD và BD ⇒ D (3; 1)

Trung đim I ca BD có ta đ I

1;

5

2 ⇒ C (1;0)

Vy B (−1;4); D (3;1); C (1;0)


Đ bài 10 : Trong h trc ta đ Oxy cho hình thoi ABCD có phương trình đưng chéo BDlà (d) : x − y = 0. Đưng thng AB đi qua đim P 1; √ 3, đưng thng CD đi qua đimQ −2; −2√ 3. Tìm ta đ các đnh ca hình thoi bit đ dài AB = AC và đim B có hoành đln hơn 1.

Li gii tham kho :

Ta có AB = AC ⇒ tam giác AB C đu ⇒ ABC = 60o ⇒ ABD = 30o

Đưng thng B D có vtpt −→n1 = (1; −1). Gi s −→n = (a; b) là vtpt ca AB

⇒ cos (−→n1, −→n ) = |a − b|√ 2.

√ a2 + b2

=

√ 3

2 ⇔ a2 + 4ab + b2 = 0 ⇔ a = −2 ± √ 3 b

Vi a = −2 −√

3

b chn −→n = −2 −√

3; 1

đưng thng AB đi qua đim P và có vtpt −→n ⇒AB : 2 + √ 3x − y − 2 = 0Ta đ đim B là giao đim ca AB và B D ⇒ B

2

1 +√

3;

2

1 +√

3

loi do xB > 1

Vi a = −2 + √ 3 b chn −→n = −2 + √ 3; 1 đưng thng AB đi qua đim P và có vtpt −→n ⇒AB : 2 − √ 3x − y − 2 + 2√ 3 = 0Ta đ đim B là giao đim ca AB và B D ⇒ B (2;2) tha mãn

Ta có C D // AB và C D đi qua đim Q

⇒CD : 2

−

√ 3x −

y + 4

−4√

3 = 0

Ta đ đim D là giao đim ca B D và C D ⇒ D (−4; −4) ⇒ ta đ tâm k ca hình thoi là trung đimca BD ⇒ K (−1; −1)


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


10/42

w w w

. d e t

i t u . n

e t


Phương trình đưng chéo AC đi qua đim K và vuông góc vi B D ⇒ AC : x + y + 2 = 0

Ta đ đim A là giao đim ca AB và AC ⇒ A (....)

Ta đ đim C là giao đim ca C D và AC ⇒

C (....)


Đ bài 11 : Trong h trc ta đ Oxy cho tam giác ABC có A (5;2)phương trình đưng trung trccnh BC và trung tuyn xut phát t đnh C ln lưt là (d1) : 2x+y−5 = 0; (d2) : x+y−6 = 0.Tìmta đ các đnh B , C ca tam giác AB C .

Li gii tham kho :

Gi s đim B (a; b). Ta có trung đim ca AB là M

a + 52

; b + 22

∈ (d2)⇒ a + 5

2 +

b + 2

2 − 6 = 0 ⇔ a + b − 7 = 0 ⇔ b = 7 − a ⇒ B (a; 7 − a)

Ly đim C (c; 6 − c) ∈ (d2)

(d1) là trung trc ca B C ⇒ trung đim ca B C là N

a + c

2 ;

13 − a − c2

∈ (d1)

⇒ a + c + 13 − a − c2

− 5 = 0 ⇔ a + c (1)+ 3 = 0

(d1) là trung trc ca B C ⇒ BC ⊥(d1) ⇒ −−−→BC ⊥−→ud1 ta có −→ud1 = (1; −2) ; −−−→BC = (c − a; a − 1 − c)

⇒ c − a − 2 (a − 1 − c) = 0 ⇔ 3c − 3a (2)+ 2 = 0

T (1) và (2) ta có

c + a = −33c − 3a = −2

⇔

a = −76

c = −116

⇒ ta đ đim B và C


Đ bài 12 : Trong h trc ta đ Oxy cho tam giác ABC có A (−1; −3), trc tâm H (1; −1) vàtâm đưng tròn ngoi tip tam giác I (2; −2). Xác đnh ta đ các đnh B , C ca tam giác AB C .

Li gii tham kho :Gi D là đim đi xng vi A qua I ⇒ AD là đưng kính đưng tròn tâm I và I là trung đim caAD ⇒ D (5; −1)

AD là đưng kính đưng tròn tâm I ⇒

CD

⊥AC , H là trc tâm

⇒BH

⊥AC

⇒CD//BH

Tương t ta có CH//BD ⇒ BHCD là hình bình hành ⇒ BC và DH ct nhau ti trung đim mi đưng


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


11/42

w w w

. d e t

i t u . n

e t


⇒ trung đim M ca DH là trung đim ca BC ta có M (3; −1)

Phương trình đưng thng BC đi qua đim M và vuông góc vi AH ⇒ BC : x + y − 2 = 0

Phương trình đưng tròn tâm I có bán kính IA = √ 10

⇒ (C ) : (x − 2)2 + (y + 2)2 = 10

Ta đ đim B và C là giao đim ca đưng thng BC và (C )


Đ bài 13 : Trong h trc ta đ Oxy cho tam giác ABC có đưng cao BH : x + 2y − 3 = 0, trungtuyn AM : 3x + 3y − 8 = 0. Cnh BC đi qua đim N (3; −2). Tìm ta đ các đnh B, C ca tamgiác ABC bit đnh C thuc đưng thng (d) : x

−y + 2 = 0.

Li gii tham kho :

Ly đim B (3 − 2b; b) ∈ BH và C (c; c + 2) ∈ (d)

Gi M là trung đim ca BC ⇒ M

3 − 2b + c2

; b + c + 2

2

. Ta có M ∈ AM

⇒ 3. 3 − 2b + c2

+ 3.b + c + 2

2 − 8 = 0 ⇔ 3b − 6c (1)+ 1 = 0

Cnh BC đi qua đim N (3; −2) ⇒ −−−→BN và −−−→N C cùng phươngTa có

−−−→BN = (2b; −2 − b) và −−−→N C = (c − 3; c + 4)

⇒ c − 32b

= c + 4

−2 − b ⇔ 3bc + 5b + 2c − (2)6 = 0

T (1) và (2) ⇒ b = ...; c = ... ⇒ ta đ đim B và C.


Đ bài 14 : Trong h trc ta đ Oxy cho hình thang cân ABCD vi C D = 2AB, phương trình

hai đưng chéo AC và BD ln lưt là (d1) : x + y − 4 = 0; (d2) : x − y − 2 = 0. Bit rng ta đ haiđim A và B đu dương và din tích hình thang bng 36. Tìm ta đ các đnh hình thang.


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


12/42


13/42

w w w

. d e t

i t u . n

e t


Gi s B (a; b) có AB⊥BC ⇒ −−−→AB⊥−→AC có −−−→AB = (a − 1; b − 5) ; −−−→BC = (a − 2; b + 3)

⇒ (a − 1) (a − 2) + (b − 5) (b (1)+ 3) = 0

Ta có C M = 2BC ⇒ −−→CM = 2−−−→BC ⇒ M (6 − 2a; −9 − 2b)

M N ⊥BN ⇒ −−→M N ⊥−−−→BN mà −−−→BN =

a + 5

2; b − 1

2

;−−→M N =

17

2 − 2a; −19

2 − 2b

⇒

a + 5

2

17

2 − 2a

+

b − 1

2

−19

2 − 2b

(2)= 0

T (1) và (2) ⇒ a = ....; b = .... ⇒ B (....)


Đ bài 16 : Trong h trc ta đ Oxy cho hình ch nht ABCD, bit phân giác trong góc ABC đi qua trung đim M ca cnh AD, phương trình đưng thng BM là (d) : x − y + 2 = 0, đimD thuc đưng thng (d1) : x + y − 9 = 0, điêm E (−1;2) thuc đưng thng AB và đim B cóhoành đ âm. Tìm ta đ các đnh ca hình ch nht

Li gii tham kho :

Ta có B M là phân giác góc ABC ⇒ ABM = 45o ⇒ ∆ABM vuông cân ti A

Gi −→n = (a; b) là vtpt ca đưng thng AB , có −→n1 = (1; −1) là vtpt ca BM

⇒ cos(−→n , −→n1) = |a − b|√ 2.

√ a2 + b2

=√

2

2 ⇔ ab = 0 ⇔

a = 0

b = 0


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


14/42

w w w

. d e t

i t u . n

e t


Vi a = 0 chn −→n = (0; 1) ⇒ phương trình đưng thng AB đi qua đim E và có vtpt −→n ⇒ AB :y − 2 = 0 ⇒ Ta đ B là giao đim ca AB và B M ⇒ B (0;2) ( loi)

Vi b = 0 chn −→n = (1;0) ⇒ phương trình đưng thng AB đi qua đim E và có vtpt −→n ⇒ AB :x + 1 = 0 ⇒ Ta đ B là giao đim ca AB và B M ⇒ B (−1;1) ( tha mãn)

Gi s đim A (−1; a) ∈ AB và D (d; 9 − d) ∈ (d1)

Trung đim M ca AD có ta đ M

d − 12

; 9 − d + a

2

∈ (d)

⇒ d − 12

− 9 − d + a2

+ 2 = 0 ⇔ 2d − a − (1)6 = 0

Ta có AD⊥AB ⇒ −−−→AD⊥−−−→AB mà −−−→AB = (0; 1) và −−−→AD = (d + 1; 9 − d − a)

⇒9

−d

−a = 0

⇔a + d (2)= 9

T (1) và (2) ⇒

d = 5

a = 4⇒

A (−1;4)D (5;4)

Gi I là tâm hình ch nht ⇒ I

2; 5

2

. I là trung đim ca AC ⇒ C (5; 1)


Đ bài 17 : Trong h trc ta đ Oxy cho tam giác ABC vuông ti A, bit B và C đi xng nhau

qua gc ta đ O. Đưng phân giác trong góc B có phương trình (d) : x + 2y − 5 = 0. Tìm ta đcác đnh ca tam giác AB C bit đưng thng AC đi qua đim K (6; 2).

Li gii tham kho :

Gi đim B (5 − 2b; b) ∈ (d). B và C đi xng nhau qua gc ta đ O ⇒ C (2b − 5; −b)

Qua O k đưng thng vuông góc vi (d) ct (d) và AB ln lưt ti F và I .

Đưng thng OF đi qua O và vuông góc vi (d) ⇒ OF : 2x − y = 0

Ta đ F là giao đim ca (d) và OF ⇒ F (1; 2)

F là trung đim ca OI ⇒ I (2; 4)


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


15/42

w w w

. d e t

i t u . n

e t


−−−→ −→ −−−→ −→Tam giác AB C vuông ti A ⇒ AB⊥AC ⇒ AB⊥AC có AB = (3 − 2b; b − 4) và AC = (2b − 11; −b − 2)

⇒ (3 − 2b) (2b − 11) + (b − 4) (−b − 2) = 0 ⇔ −5b2 + 30b − 25 = 0 ⇔

b = 1

b = 5Vi b = 1 ⇒ B (3; 1) ⇒

C (−3; −1)

Phương trình đưng thng AB đi qua B và I ⇒ AB : 3x + y − 10 = 0

Phương trình đưng thng AC đi qua C và K ⇒ AC : x − 3y = 0

A là giao đim ca AB và AC ⇒ A (3; 1) ( loi do trùng đim B )

Trưng hp b = 5 xét tương t


Đ bài 18 : Trong h trc ta đ Oxy cho hình thang ABCD có din tích bng 458

. Phương trình

hai cnh đáy AB : x − 3y + 1 = 0 và CD : 2x − 6y + 17 = 0. AD và BC ct nhau ti đim K (2;6).Hai đưng chéo ct nhau ti đim I

1;

7

3

. Xác đnh ta đ các đnh ca hình thang ABCD.

Li gii tham kho :

Khong cách gia AB và CD là d = 15√

40

Ta có din tích hình thang S = 1

2. (AB + CD) .d ⇒ AB + CD = 3

√ 10

2 (1)

ABCD là hình thang ⇒ ABCD

= d (I,AB)

d (I , C D) (2)= 2

T (1) và (2) ⇒ AB = 2.CD = √ 10

Tam giác KAB có CD // AB và AB = 2CD ⇒ CD là đưng trung bình ca tam giác KABNi KI ct AB và CD ti M và N ⇒ M. N ln lưt là trung đim ca AB và CD


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


16/42

w w w

. d e t

i t u . n

e t


Phương trình đưng thăng KI đi qua K và I ⇒ KI : 11x − 3y − 4 = 0

M là giao đim ca KI và AB ⇒ M

1

2; 1

2

Ta có AB = √ 10 và M là trung đim ca AB ⇒ A và B thuc đưng tròn tâm M bán kính R = √ 102

⇒ (C ) :

x − 12

2

+

y − 1

2

2

= 5

2

A, B là giao đim ca (C ) và đưng thng AB ⇒ A, B có ta đ là (2; 1) ; (−1;0)

Do đó C, D có ta đ là

2; 7

2

;

1

2; 3


Đ bài 19 : Trong h trc ta đ Oxy cho tam giác ABC có BC = 2AB, phương trình đưngtrung tuyn xut phát t đnh B là (d) : x + y − 2 = 0. Bit ABC = 120o và đim A (3;1). Tìmta đ các đnh còn li ca tam giác.

Li gii tham kho :

Đt AB = x ⇒ BC = 2x. Áp dng đnh lý Cosin vào tam giác AB C ta có

AC 2 = AB2 + BC 2 − 2.AB.BC. cos ABC = 7x2 ⇒ AC = x√ 7

Áp dng công thc tính đưng trung tuyn vào tam giác AB C ta đưc

BM 2

=

AB2 + BC 2

2 − AC 2

4 =

3x2

4

Trong tam giác AB M có AB = x, BM 2 = 3x2

4 ; AM =

x√

7

2 ⇒ AM 2 = AB2 + BM 2

⇒ ∆ABM vuông ti B ⇒ AB⊥BM

Phương trình đưng thng AB đi qua A và vuông góc vi B M ⇒ AB : x − y − 2 = 0

B là giao đim ca AB và B M ⇒ B (2;0)

Li có AB = d (A,BM ) =√

2 = x

⇒BM =

√ 6

2

. Gi M (m; 2−

m)

∈BM

⇒ BM 2 = 2 (m − 2)2 = 32 ⇔ m = 2 ±

√ 3

2


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


17/42


18/42

w w w

. d e t

i t u . n

e t


Đ bài 21 : Trong mt phng vi h ta đ Oxy cho tam giác ABC có đim A (−1; −3) , B (5;1).Đim M nm trên đon thng BC sao cho M C = 2M B. Tìm ta đ đim C bit rng M A =

AC = 5 và đưng thng B C có h s góc là mt s nguyên.

Li gii tham kho :

Gi s đim M (a; b) ta có M A = 5 ⇒ (a + 1)2 + (b + 3)2 = 25

a2 + 2a + b2 + 6b (1)= 15

Gi D là trung đim ca C M ta có M A = AC = 5 ⇒ ∆CAM cân ti A ⇒ AD⊥CM

Theo gi thit M C = 2M B ⇒ M B = M D ⇒ M là trung đim ca B D ⇒ D (2a − 5; 2b − 1)−−−→AD = (2a

−4; 2b + 2) ;

−→BI = (2a

−10;2a

−2)

AD⊥BI ⇒ −−−→AD.−→BI = 0 ⇒ (2a − 4)(2a − 10) + (2b + 2) (2b − 2) = 0

⇒ a2 − 7a + b2 = − (2)9

T (1) và (2) ⇒

a = 2; b = 1

a = 50

13; b = −23

13

Vi a = 50

13; b = −23

13 ⇒ M

50

13; −23

13

Phương trình đưng thng BC đi qua B và M ⇒ BC : 12x − 5y − 55 = 0 ( loi do phương trìnhBC có h s góc nguyên)

Vi a = 2; b = 1 ⇒ M (2; 1) phương trình B C đi qua M và B ⇒ BC : y = 1 ( tha mãn)

Ta đ đim D (−1;1) ⇒ C (−4;1)


Đ bài 22 : Trong mt phng vi h ta đ Oxy cho tam giác ABC cân ti A, có trc tâm H (−3;2).

Gi D, E là chân đưng cao h t B và C . Đim A thuc đưng thng (d) : x − 3y − 3 = 0, đimF (−2;3) thuc đưng thng DE và H D = 2. Tìm ta đ đnh A.


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/


19/42

w w w

. d e t

i t u . n

e t

www.dethithu.net HÌNH HC GII TÍCH TRONG MT PHNG

Li gii tham kho :

Ta có H D = 2 ⇒ (xD + 3)2

+ (yD − 2)2 = 4⇔ x2

D + y2

D + 6xD − 4yD + 9 = 0 (1)

Đim A ∈ (d) ⇒ A (3a + 3; a) ta có AD⊥DH ⇒ −−−→AD. −−−→HD = 0

(xD − 3a − 3) (xD + 3) + (yD − a) (yD − 2) = 0

x2D

+ y2D

− 3axD − (a + 2) yD − 7a − 9 = 0 (2)

T (1) và (2) ⇒ (6 + 3a) xD + (a − 2) yD + 7a + 18 = 0

Tương t ta có (6 + 3a) xE + (a − 2) yE + 7a + 18 = 0Do đó phương trình đưng thng DE có dng (d1) : (6 + 3a) x + (a − 2) y + 7a + 18 = 0

Mà đim F ∈ (d1) ⇒ a = 0 ⇒ A (3; 0)

Đ bài 23 : Trong mt phng vi h ta đ Oxy cho tam giác ABC có trng tâm G (1;1),đưng cao t đnh A có phương trình (d) : 2x − y + 1 = 0. Các đnh B và C thuc đưng thng(d1) : x + 2y − 1 = 0. Xác đnh ta đ các đnh ca tam giác bit tam giác ABC có din tích bng6.

Li gii tham kho :

Đim A ∈ (d) ⇒ A (a; 2a + 1)


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://facebook.com/groups/onthidhtoananhvanhttp://dethithu.net/http://www.dethithu.net/http://www.dethithu.net/


20/42

w w w

. d e t

i t u . n

e t

www.dethithu.net HÌNH HC GII TÍCH TRONG MT PHNG

Gi M là trung đim ca B C ⇒ G ∈ AM và AG = 2GM ⇒ −→AG = 2−−→GM

⇒ M

3 − a2

; 1 − a

mt khác M ∈ (d1)

⇒ 3 − a2

+ 2 (1 − a) − 1 = 0 ⇒ a = 1 ⇒ A (1;3) ⇒ M (1; 0)

Gi H là giao đim ca (d) và (d1) ⇒ H

−15

; 3

5

⇒ AH = 6√

5

S = 1

2.AH.BC = 6 ⇒ BC = 2√ 5 ⇒ M B = M C = √ 5

Đim B ∈ (d1) ⇒ B (1 − 2b; b) ⇒ M B2 = 5b2 = 5 ⇔ b = ±1 b = 1 ⇒ B (−1;1) ⇒ C (3; −1)

b = −1 ⇒ B (3; −1) ⇒ C (−1;1)Bài toán gii quyt xong.

Đ bài 24 : Trong mt phng vi h ta đ Oxy cho hình ch nht ABCD có din tích bng 6.Phương trình đưng thng cha đưng chéo B D là (d) : 2x + y − 11 = 0, đưng thng AB đi quađim M (4; 2), đưng thng B C đi qua đim N (8; 4). Xác đnh ta đ các đnh ca hình ch nht

bit các đim B , D đu có hoành đ ln hơn 4.

Li gii tham kho :

Vì B ∈ (d) ⇒ B (b; 11 − 2b). AB⊥BC ⇒ M B⊥N B ⇒ −−→M B. −−−→N B = 0

⇒ (b − 4) (b − 8) + (9 − 2b) (7 − 2b) = 0 ⇒ 5b2 − 44b + 95 = 0 ⇔ b = 195

b = 5⇒ B (5; 1)

Phương trình đưng thng AB đi qua đim B và M ⇒ AB : x + y − 6 = 0

Phương trình đưng thng B C đi qua đim B và N ⇒ AC : x − y − 4 = 0

A ∈ AB ⇒ A (a; 6 − a) và C ∈ BC ⇒ C (c; c − 4)

Gi I là tâm ca hình ch nht ⇒ I

a + c

2 ;

c − a + 22

∈ BD

⇒ a + c + c − a + 22

− 11 = 0 ⇔ 3c + a − 20 = 0 (1)

AB =√

2. |a − 5| và B C = √ 2. |c − 5| ⇒ S = 2 |a − 5| . |c − 5| = 6 (2)


http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethithu.net/http://www.dethith

Documents

[DeThiThu.Net]50bai-oxy-loi-giai-chi-tiet.pdf