Phương pháp giải toán hình học theo chuyên đề

T R U N G T A M LUYCN T H I D A I HOC V I N H VI£N SAI G O N
Tdng chu bi§n: PHAM H 6 N G D A N H
NGUYEN PHU KHANH - N G U Y I N TAT THU
NGUYEN TAN SIENG - TRAN VAN TOAN - NGUYEN ANH TRUCfNG

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(Nhdm giao vien chuyen luyen thi B^i hpc)

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PHUONG PHAP GIAI TOAN

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theo chuyen de
H I N H HOC T R O N G K H O N G G I A N
*

H I N H HOC T Q A OO T R O N G K H O N G G I A N

H I N H HOC T O A OO T R O N G M A T P H A N G

THU VIEN l\m 8 I N H THUAN]

N H A X U A T B A N D A I HOC

QU6c

GIAHA NQI

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Ctij TNHH

N H f l X U R T B R N D f l l H O C Q U O C G l f l Ht{

MTV

DVVH

Khang

Viet

NOI

16 Hang Chuoi - Hai Ba TrUng - Ha Npi
Dien thoai : Bien t a p - Che ban: (04) 39714896;
Hanh chinh: (04) 39714899; Tona bien t a p : (04) 39714897

Fax: (04) 39714899

P H U O N G P H A P T O A D O T R O N G IVIAT P H A N G
A , LY THUYET G I A O K H O A
I. Tpa dp trong mat phang.

Chiu trdch nhiem xuat ban

01

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• Cho u ( x p y j ) ; v(x2;y2) va k e R . K h i do:
1) u + v = (xi + X 2 ; y i + y 2 )

oc

- Tong bi&n tap :

TS. P H A M T H j T R A M

Che

:

C O N G TY KHANG V I E T

:

C O N G TY KHANG V I E T

bay bia

u=vc^r^

6) U . V = X ] X 2 + y ] y 2 = > u l v < ; : > u . v = 0<=> \-^\2 + y ] y 2 = 0
•

Haivecta

u ( x j , y j ) ; v ( x 2 ; y 2 ) c i i n g p h i r a n g v a i n h a u <=>

ie

Trinh

Z=Jx\+y\)

hi

N G Q C LAM

"''""''^

2) u - v = ( x i - X 2 ; y i - y 2 )

nT

:

uO

Bien tap
ban

4)

Da

3) k u = ( k x i ; k y i )

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Gidm doc

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• Goc g i i j a hai vec to u ( x j , y j ) ; v ( x 2 ; y 2 ) :

Ta

Tong phdt hanh va doi tdc lien ket xuat ban:

s/
CONG TYTNHH

lpiS|r

MTV

PHLfONG PHAP GIAI TOAN
M a so:

HINH H Q C

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SACK L I E N K E T

2) ^^=^3

= ^{x^ - x

+ {y ^ - y

_ X A + X B
I ~

3)

t r o n g d o I la t r u n g d i e m ciia A B .

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Website: www.nhasachkhangvlet.vn

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Email:

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1) A B = ( x B - X A ; y B - y A )

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f D i a c h I 71 Dinh Tien Hoang - P Da Kao - Q 1 - TP HCN/I
^
Dien thoai: 0873911569^^ 39105797 - 39111969 - 39111968
Fax: 08. 3911 0880

XiX2+yiy2

Cho A ( x ^ ; y ^ ) ; B ( x B ; y B ) . K h i do :

ro

DjCH Vg VAN HOA KHANG V I E T

U.V

u

up

M^^k

^

cos(u,v)=

THEO CHUYEN DE

•

AB 1 CD o

AB.CD - 0

• Cho tarn giac A B C v o i A{x^;y^),

G ( x ( , ; y g ) ciia tarn giac A B C la :

1L-321DH2012

In 2.000 c u o n , kho 1 6 x 2 4 c m
T a i : Cty T N H H MTV IN A N MAI T H j N H DLfC

B(xB;yB), C{x^;y^).

K h i d o t r o n g tarn

V
_ X A + X B + X C
X G ^

yG=

I

I I . PhirotTg trinh duong thang

Dja chi: 7 1 , Klia Van Can, P. H i e p Binh C h a n h , Q . Thu Dufc, TP. Ho Chi M i n h

1. 'Phuang trinh duong thdng

So xuat bSn: 1335 - 201 2/CXB/07 - 21 5 / D H Q G H N ngay 0 6 / 1 1 / 2 0 1 2 .

1.1. Vec to chi phucmg (VTCP), vec to phdp tuyen (VTPT) cua duong

thang:

Cho d u o n g t h a n g d .

Quyet d i n h xuat b5n so: 3 1 8 L K - T N / Q D - N X B D H Q G H N , cap ngay 12/11/2012
In xong va nop luu chieu Q u y I n a m 201 3

,, ,^,

•

n = (a;b) ?t 0 g o i la vec t o p h a p t u y e n cua d neu gia ciia no v u o n g v o i d .

3


Phiam^ phiip giui loiin llinli hoc Iheo chuycn de- Nguyen Phti Khdnh, Nguyen Tat Thti

u = ( u j ; u 2 ) ^ 0 goi la vec ta chi phuong cua d ne'u gia cua no trung hoac
d(M,(A)):

song song voi duong thang d.

axp + byp + c
Va^+b^

Mot duong thang c6 v6 so VTPT va v6 so VTCP ( Cac vec to nay luon cung

5. (phuong trinh duang phdn gidc cua goc tao boi hai duang thdng

phuong voi nhau)

Cho hai duong thang d^ : a^x + b^y + c^ = 0 va d2 : a j X + b2y + Cj = 0

• Moi quan he giua VTPT va VTCP: n.u = 0 .

Phuong trinh phan giac ciia goc tao boi hai duong thang la: - , v , •
ajX

cua duong thang d .
~!( 5

f

oc

•

+ b^y + Cj

/

• Ne'u n = (a; b) la mpt VTPT cua duong thang d thi u = (b; -a) la mot VTCP

01

•

Cty TNHH MTV DWH Khang Viet

+

III. Phuang trinh duong tron.

1.2. Phuwig trinh dumig thang
1.2.1. Phuatig trinh tong qudt cua duong thang:

1.
^/a^^b[

•

iH

• Duong thang AB c6 AB la VTCP.

a2X + b2y + C2
, i c-i..; ;

hi

Da

rmu.j.

Cho duong tron (C) tam I(a; b ) , ban kinh R, khi do phuang trinh ciia (C)

nT

Cho duong thMng d di qua diem A(xQ;yQ) va c6 n = (a;b) la VTPT, khi do

la: ( x - a ) 2 + ( y - b ) 2 = R 2 .

uO

phuong trinh tong quat ciia d c6 dang: a(x - X Q ) + b(y - yp) = 0.

Ngoai ra phuong trinh: x^ + y ^ - 2 a x - 2 b y + c = 0 voi a ^ + b ^ - o O

Cho duong thSng d di qua diem A(xo;yo) va c6 u = (a;b) la VTCP, khi do

Ta

XQ

la phuong trinh ciia duong tron c6 tam I(a;b), ban kinh R = Va^ + b^ - c .

+ at

2. Phuang trinh tiep tuyen:

s/

X =

phuong trinh tham so cua duong thang d la:

, t G R.

up

[y = y(,+bt

ro

2. Vi tri tuang doi giua hai duang thdng.

Cho duong tron (C): ( x - a ) ^ + ( y - b ) ^ = R ^
• Tiep tuyen A ciia (C) tai diem M la duong thang d i qua M va vuong goc
vai I M .

|a,x + b,y + Cj = 0

tuong doi giua chung phu thuoc vao so nghiem cua h^ : <
,
(I)

• Duong thang A : Ax + By + C = 0 la tiep tuyen ciia (C) <=> d(I, A) = R '

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Cho hai duong thcing dj : a^x + bjy + c^ = 0; d2 : a2X + b2y + C2 = 0 . Khi do vi tri

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[a2X + b 2 y + C2 =0

• Neu (I) v6 nghiem thi d^ / /d2 .

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• Ne'u (I) v6 so nghiem thi d^ = d j

Cho hai duong thang

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Ne'u (I) CO nghiem duy nha't thi dj va d2 cat nhau va nghiem ciia he la toa

do giao diem.
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3. Goc giua hai dijcang thdng.

dj : a j X + b^y+ Cj =0; d2 :a2X + b2y + C2 = 0 . Goi a

la goc nhon tao boi hai duong thang dj va d2 .
Ta CO : cosa =

aja2 + bjb2

^/a^Tb^ ^/af+b
4. JChodng each tit mot diem den ducrng thdng.
Cho duong th5ng A : ax + by + c = 0 va diem M ( X Q ; y ^ ) . Khi do khoang each
tu M den A dugc tinh boi cong thuc:

cQng

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1.2.2. Phuovg trinh tham so cua duong thang:

•

.

• Duong tron (C): (x -a)^ + (y - b)^ = R^ c6 hai tiep tuyen cung phuong voi
Oy la x = a ± R . Ngoai hai tiep tuyen nay cac tiep tuyen con lai deu c6 dang:
y = kx + m .
IV. E lip
1. 'i)inh nghra.-Trong mat phang cho hai diem co'djnh Fi,F2 c6 Y^Yj =2c. Tap
hop cac diem M cua mat phang sao cho MF^ +MF2 =2a (2a khong doi va
a > c> 0) la mot duong elip.
• F,,F2 : la hai tieu diem va 2c la tieu cu ciia elip.
•

MF|,MF2 : la cac ban kinh qua tieu.

2. Phuang trinh chinh tdc cua elip:
4 +4
a 2 b^
Vay diem M(xo;y(,) e (E) •

= ^ voi b^=a^-c^.

= 1 va


K'.

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Cty TNHH MTV DWH Khang Viet

Phumtg phcip giiii Toan Hhih hoc theo chuyen tie- Nguyen Phu Khdnh, Nguyen Tat Thu

• D O dai cac ban kinh qua tieu cua M ( x o ; y ( , ) e ( H ) :
3. Tinh chat v>d hlnh dang cua elip: Cho (E): — + ^
a
b

= 1, a > b .

• True doi xung Ox,Oy . Tarn do'i xiing O .

+) MF^ = ex„ + a va MF2 = e X ( , - a khi

+) MFj = -exp - a va MF2 = -exp + a khi

j,

• Dinh: A[(-a;0), A2(a;0), 6^(0;-b) va 62(0; b ) . A^A2 = 2a goi la do dai

2

true Ion, B]B2 = 2b goi la do dai true be.

0.

2

b

/

a

01

j.<^inhnghia:

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Parabol la tap hop cae diem M cua mat phang each deu mot duong thang

Da

A co'dinhvamot diem F co dinh khong thuoe A .

0

<1

R

Bi

2. 'Phuxmg trinh chinh tdc cua ^arabd:

= 2px

uO

a

a^
Hai duong chuan: X = ± — = ± —
e
e

nT

s

hi

A : duong chuan; F : tieu diem va d(F,A) = p > 0 la tham so'tieu.

a

3.jrinh dang cua Parabol (
ie

•

Tam sai: e = — =
a

VI. Parabol

Q

XQ <

oc

^

.2

D

a

• Noi tiep trong hinh ehir nhat co so PQRS
CO ki'ch thuoc 2a va 2b voi b^ = a^ - e^.
•

..2

0.

M ( x o ; y o ) 6 ( H ) : \ - J ^ = l « ^ - f j = l vataluonco X(,]>a.

.

• Tieu diem: F|(-c;0), F2(e;0).

XQ >

iL

-ex(,.

• True Ox la true do'i xung, dinh O. Tieu diem F ( ^ ; 0 ) .
• Duong chuan A : x =

ro

up

FjF2 =2c. Tap hop cac diem M ciia mat phSng sao eho MF^ - M F j =2a (2a

s/

1. ^inh nghia: Trong mat phang voi h$ toa do Oxy eho hai diem Fi, F2 eo

Ta

V. Hypebol

/g

khong doi va c > a > 0 ) la mpt Hypebol.

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• Fp F2 : la 2 tieu diem va F|F2 = 2e la tieu eu.

3. Tinh chat vd hlnh dang cua hypebol (fi):

y^

= 1 voi

h^=c^-a^.

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ce

x^
2. 'Phimng trinh chinh idc cua hypebok
a^

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• 1VIF[,MF2 : la eac ban kinh qua tieu.

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• True doi xung Ox (true thuc), Oy (true ao). Tam doi xung O .
• Dinh: Aj(-a;0), A2 (a;0). D Q dai true thuc: 2a va do dai true ao: 2b.
• Tieu diem Fi(-e; 0), Fj ( c; O) .

•

M ( x ; y ) e ( P ) : MF = x + ^ voi x > 0 .

B, CAC BAI THlfONG GAP
§ 1.

cAc

BAI T O A N C O B A N

1. Xg.p phuang trinh duang thang.
De lap phuong trinh duong thang A ta thuong dung cac each sau
• Tim diemM(xo;yo) ma A di qua va mot VTPT n = (a;b). Khi do phuong
trinh duong thang can lap la: a(x - XQ) + b ( y - yp) = 0 .
• Gia su duong thang can lap A : ax + by + e = 0 . Dua vao dieu kien bai toan ta
tim dugc a = mb,c = nb. Khi do phuong trinh A : m x + y + n = 0. Phuong phap

•

Hai tiem can: y = ± —x
a

•

Hinh eho nhat co so PQRS c6 kieh thuoe 2a, 2b voi b^ = c^ - a^.

nay ta thuong ap dung doi voi bai toan lien quan den khoang each va goe
• Phuong phap quy tich: M(xQ;yQ)e A:ax + by + e=^Oc:> axy + by^ + e = 0 .
Vidu 1.1.1.Trong mat phSng voi he toa do Oxy cho duong tron

• Tam sai: e = — =
a

(C):(x-])2+(y-2)2=25.
1) Viet phuong trinh tiep tuyen ciia (C) tai diem M(4;6), '

•

Hai duong chuan: x = ±— = ± —

2) Viet phuong trinh tiep tuyen cua (C) xua't phat tu diem N ( - 6 ; l )


Cty TNHH MTV DWH

Phucntg phap giai ToAn Ilinh hoc theo chuycn lic- Nguyen Pliii Khanh, Nguyen Tat Thii

Khang Viet

D u a vao gia thie't cua bai toan ta t i m dugc a , b , c . Cach nay ta t h u o n g ap

3) T u E(-6;3) ve hai tie'p tuye'n EA, EB (A, B la tie'p diem) den (C). Viet

d u n g k h i yeu cau viet p h u o n g t r i n h d u o n g tron d i qua ba d i e m .

p h u o n g t r i n h d u o n g thang A B .

Vi du 1.1.2. Lap p h u o n g t r i n h d u o n g tron (C), bie't
1) (C) d i qua A ( 3 ; 4 ) va cac h i n h chie'u ciia A len cac true toa do.

D u o n g tron (C) c6 tam 1(1; 2 ) , ban k i n h R = 5 .

2

1) Tie'p tuyen d i qua M va v u o n g goc v o i I M nen nhan I M = (3;4) l a m VTPT
N e n p h u o n g t r i n h tie'p tuye'n la: 3(x - 4) + 4(y - 6) = 0 <=> 3x + 4y - 36 = 0 .

01

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1) Goi A i , A2 Ian i u g t la h i n h chie'u ciia A len hai true Ox, O y , suy ra

(*)

Da

A : a ( x + 6) + b ( y - l ) = 0<=>ax + by + 6 a - b = 0, a^ + b^

A,(3;0), A2(0;4).

hi

Ta c6:
7a + b =

5

^

o{7a

+ b)^ = 2 5 ( 3 ^

nT

G i a s i i ( C ) : x ^ + y ^ - 2 a x - 2 b y + e = 0.
•=5o

+b^)

uO

7a+ b

•

4
thay vao n ta c6: — b x + by - 9b = 0 «• 4x - 3y + 27 = 0 .
3

<=>

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w.

lA.NA = 0

=25

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Ta

fa

ce

3x + 4y +14 = 0 va 4x - 3y + 27 = 0.

[(a - l)(a + 6) + (b - 2)(b - 3) = 0

= ^ 7 a - b + 20 = 0

2) Goi I(a;b) la t a m ciia d u o n g t r o n (C), v i l € ( C i ) nen: ( a - 2 )

7

7

+b

4

=-

Do (C) tie'p xuc v o i hai d u o n g t h i n g A ^ A j nen d ( I , A j ) = d ( I , A2)
a-b

a-7b

V2

5V2

• b = -2a

<=>b = -2a,a = 2b

thay vao (1) ta CO dugc:

(a - if- + 4a^ = - <=> 5a^ - 4a + — = 0 p h u o n g t r i n h nay v 6 n g h i e m

a^ + b^ + 5 a - 5 b = 0

T u o n g t u ta cung c6 dug-c B e A = > A B = A = > A B : 7 x - y + 20 = 0 .
2. Cdch lap phimng trinh dizcrng tron.
De lap p h u o n g t r i n h d u o n g t r o n (C) ta t h u o n g su d u n g cac each sau
Cdch 7 ; T i m tam I(a;b) va ban k i n h ciia d u o n g t r o n . K h i do p h u o n g t r i n h
.

Cdch 2 ; G i a su p h u o n g t r i n h d u o n g tron co dang: x^ + y^ - 2ax - 2by + c = 0
8

e= 0

Vay p h u o n g t r i n h (C): x^ + y^ - 3x - 4y = 0 .

a^ + b^ - 2 a - 4 b - 2 0 = 0

T u do ta suy ra duoc A e A : 7 x - y + 20 = 0.

d u o n g tron co dang: (x - a ) ^ + ( y - b)^ =

- 8 b + e = -16

3
a =—
2
<=> •!b = 2 .

s/

bo
ok

Vay CO hai tie'p tuye'n thoa yeu cau bai toan la:
3) Goi A ( a ; b ) . T a c 6 :
Ae(C)
(a-1)^ + ( b - 2 ) ^

ro

3
7
thay vao (*) ta c6: - b x + by + - b = 0 o 3 x + 4 y + 14 = 0.

/g

4
a =—b
3

+ 1 2 - - 2 4 = 0c^
b
a=-lb'

3

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om

•

-

-6a + c = - 9

Do A , A p A 2 e ( C ) nen ta co he:

a = ^b
4

up

o24a2+14ab-24b2 = 0 o 2 4

- 6 a - 8 b + e = -25

ie

Va^ + b^

3
a=-b

va tiep xiic v o i hai

d u o n g thc^ng A, : x - y = 0 va A2 : xXffigidi.
- 7 y = 0.

D o A d i qua N nen p h u o n g trinh c6 dang

R o

4
=-

/

2) (C) CO tam n a m tren d u o n g t r o n ( C j ) : (x - 2)^ + y

2) Gp i A la tie'p tuye'n can t i m .

d(I,A) =

2

9

•

4

9

4

8

-^''i-'

a = 2b thay v a o ( l ) t a c o : ( 2 b - 2 r + b ' ' = - < : : > b = - , a = - .
o
0 0
Suy ra R = D ( I , A , ) =

(

Vay p h u o n g t r i n h ( C ) : x
I
3. Cac diem, ctqc biet trong tam
Cho t a m giac A B C . K h i do:

8l

2

r

4^ '

— + y - 5 ,
5j
gidc.

8

25

-:l.:J......

(1)


CUj TNHH MTV DWH Khang Viet

Todn Hiith hoc theo chiiyen de - Nguyen Phi't Klidnh, Nguyen Tat Thu

3

'

7(x-l) + (y-3) = 0 j7x + y-10 = 0
Ma [x + 7y + 2 = 0
BH.AC = 0

3

• True tam H : AH.BC = 0
BH.AC = 0

3 1
Suy ra H 2'
2

01

lA^ = IB^
Tam duong tron ngoai tiep I: lA^ = IC^
AB.AK AC.AK
AC
• Tam duong tron noi tiep K : AB
BC.BK BA.BK
BC
AB
Chu y:C6 the tim K theo each sau:

iH

nT

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w.

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ce

1) Tim toa do true tam H, tam duong tron ngoai tiep I va trong tam G cua
tam giac ABC. Tu dp suy ra I, G, H thang hang;
2) Tim toa do tam duong tron noi tiep va tam duong tron bang tiep goc A
cua tam giac ABC.
1) Taeo

Yc

3

Goi H(x;y), suy ra

Xffigidi.

1 9

8

( 3 _21
AH = ( x - l ; y - 3 ) , B H = (x + 2;y),BC = 21 3 ,AC = 8' 8

in

r
I

x

5^ 2

8y

+

o

--]

Vy o
8j

15 31
16'16

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ie

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21 3
111^
— x + —y
^13= 13^ , GI = 13.13 >GH = -2GI. Suy ra I,G,H thang hang
Ta CO4GH =4^ 32
"l6'l6
[) Goi K{x; y) la tam duong tron noi tiep tam giac ABC. Ta c6:

Ta
s/

up

/g
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AB.AJ AC.AJ
AC
Tam duong tron bang tiep (goc A) J: AB
BJ.BC AB.BJ
BC
AB
l?jdui.i.3.Cho tam giac ABC c6 A(1;3),B(-2;0),C 5 3

ro

^

1

=

__15
16
31

hi

x+y =l

3
X=—
21
y = -:

( x - l ) 2 + ( y - 3 f =(x + 2)^+y'

oc

lA^ = IB^
Goi I(x;y), taeo: IB^ = I C ^

** Gc
Goi D la ehan duong phan giae trong goc A, ta c6: BD = AB:DC , tu day
suy ra D
AC
* Ta CO AK = AB;KD tu day ta c6 K.
BD

/

• Trong tam G

Da

Phumig phdpgidi

KAB = KAC <=>
KBC-KBA

AK,AB) = (AK,AC

COS(AK, A B ) = COS(AK, A C

BK,BA = (BK,BC)

cos (BK, B A j = COS ( B K ,

BC)

AK.AB AK.AC
AK.AB AK.AC
AK.AC
AB
AC
<=> AK.AB
BK.BA BK.BC
BK.BA BK.BC
iBK.AB ' BK.BC
I AB
BC
Ma AK = ( x - l ; y - 3 ) , B K = (x + 2;y),AB = (-3;-3) nen (*) tuong duong voi
<

-3(x-l)-3(y-3)

-8^^-^)-f^y-'^
15N/2

37^

3(x.2).3y
3V2

8^-"'^"^

2x - y = -1
x - 2 y = -2

x=0
[y = l

I5V2

8^

Vay K(0;1).
Goi J(a;b) la tam duong tron bang tiep goc A eiia tam giac ABC. Ta co:


phlip gidi Todn Hinh hoc

(A1AC

AB

2a - b = - 1

AC

2a + b = - 4

BJ.BC _ BJ.AB
BC

5,

Phu Khdnh,

AJ.AB _ AJ.AC

(B],BC) = (BJ,AB

~

AB

Nguyen

Tat

Cty TNHH

Tltu

Khang

Vie

A ( 5 ; 2 ) . P h u o n g t r i n h d u o n g t r u n g true canh BC, d u o n g t r u n g t u y e n C C
Ian l u ^ t la x + y - 6 = 0 va 2 x - y + 3 = 0 . T i m toa do cac d i n h B,C cua

tam

giac A B C .

Xgfi gidi.

4' '2.

Goi d : x + y - 6 = 0, C C : 2 x - y + 3 = 0 . Ta c6: C(c;2c + 3)
P h u o n g t r i n h BC : x - y + c + 3 = 0

oc

01

4. Cdc duang ddc hiet trong tam gidc
4.1. D u a n g t r u n g tuyen cua tam giac: K h i gap d u o n g t r u n g tuyen cua tam

Goi M la t r u n g d i e m ciia BC, suy ra M :

giac, ta chu yeu khai thac tinh chat d i qua d i n h va t r u n g d i e m cua canh do'i dien.

x +y-6 = 0

4.2. D u o n g cao cua tam giac: Ta khai thac t i n h chat d i qua d i n h va v u o n g

x-y+c+3=0

hi

goc v o i canh do'i d i e n .

y=-

2
c+ 9

uO

d i e m va v u o n g goc v o i canh do.

Suy ra B ( 3 - 2 c ; 6 - c ) = > C ' ( 4 - c ; 4 - | )

ie

4.4. D u o n g phan giac t r o n g : Ta khai thac tinh chat ne'u M thuoc A B , M ' d o i

Ta

iL

x u n g v o i M qua phan giac t r o n g goc A t h i M ' thuoc A C .

ro

A B la d i e m H ( - l ; - l ) , d u o n g phan giac t r o n g cua goc A c6 p h u o n g t r i n h

JCffigidi
K i hi?u d , : X - y + 2 = 0, d2 : 4x + 3y - 1 = 0 .

fa
ce

Goi A la d u o n g thang d i qua H va v u o n g goc v o i d j .

bo
ok

Goi H ' la d i e m d o i x u n g v o i H qua d j . K h i do H ' E A C .

.c
om

/g

x - y + 2 = 0 va d u o n g cao ke t u B c6 p h u o n g t r i n h 4x + 3y - 1 = 0 .

x + y + 2=::0

ww
w.

x-y+2=0

19.4
3 '3

, C

14

37

3'

3

5. Mot sobdi todn dung hinh ca ban.
5.1. H i n h chie'u v u o n g goc H cua d i e m A len d u o n g thang A
• L a p d u o n g thang d d i qua A va v u o n g goc v o i A
•

H=dnA

5.2. D u n g A ' d o i x u n g v o i A qua d u o n g thang A
• D u n g h i n h chie'u v u o n g goc H cua A len A
I(-2;0)

Ta CO I la t r u n g d i e m ciia H H ' nen H ' ( - 3 ; l ) .

D u o n g thang A C d i qua H ' va v u o n g goc v o i d j nen c6 p h u o n g t r i n h :
3 x - 4 y + 13 = 0 .

M a C ' e C C nen ta c6: 2 ( 4 - c ) - ( 4 - - ) + 3 = 0 < = > - - c + 7 = 0 ^ c = — .
2
2
3
Vay B

up

cua tam giac A B C bie't rang h i n h chie'u v u o n g goc cua C tren d u o n g thang

s/

Vidu 7 . i . 4 . T r o n g mat ph^ng v o i he tpa do O x y , hay xac d j n h toa do d i n h C

P h u o n g t r i n h cua A : x + y + 2 = 0 . Suy ra A n d j = I :

3-c

X = -

nT

4.3. D u o n g t r u n g true cua tam giac: Ta khai thac t i n h chat d i qua t r u n g

Lay A ' do'i x u n g v o i A qua H :

'^A'-^Xj^

x^

lyA'=2yH-yA

5.3. D u n g d u o n g t r o n ( C ) do'i x u n g v o i (C) (c6 tam I , ban k i n h R) qua d u o n g
thSng A
• D u n g r d o i x u n g v o i I qua d u o n g thang A

x-y+2=0
3 x - 4 y + 13 = 0 '

•A(5;7).

V i C H d i qua H va v u o n g v o i A H , suy ra p h u o n g t r i n h cua C H :

• D u o n g t r o n ( C ) c6 t a m I ' , ban k i n h R.
5.4. D u n g d u o n g thang d ' d o i x u n g v o i d qua d u o n g thang A .
•

[ 3 x - 4 y + 13 = 0

'if!', r<(.:

C h i i y : Giao d i e m ciia (C) va ( C ) chinh la giao d i e m cua va A .
• Lay hai d i e m M , N thuoc d . D u n g M ' , N ' Ian l u o t d o i x u n g v o i M , N qua A

3x + 4y + 7 = 0

12

DWH

Vi du 1.1.5. T r o n g mat phang v a i he toa do O x y , cho t a m giac A B C biet

5
a = —
4
b = -32

3^

Nen A C n d j = A :

MTV

/

Vay J

Nguyen

iH

(ALAB) =

theo chiiyen dc-

Da

Phuvng

3

4

d' = M ' N ' .


Phumig

phdp gidi Todii Uinh hoc theo chuyen dc - Nguyen

Vidu 1.1.6.Trong

d i e m A(3;2),

Pliii Khdnh,

Nguyen

Cty TNHH

Tat Thti

MTV DWH

Bai l - l - l - Trong mat phang Oxy cho tam giac A B C

B(-l;4).

B(4;3), C ( - 3 ; - l )

CO A ( 2 ; l ) ,

1) T i m d i e m M thuoc d u a n g thang d sao cho M A + M B nho nhat,

1) T i m toa do true t a m , t a m d u o n g t r o n ngoai tiep t a m giac A B C

2) Viet p h u o n g t r i n h d u a n g thang d ' sao cho d u o n g thang A : 3x + 4y + 1 = 0

2) Viet p h u o n g t r i n h d u o n g t r o n ngoai tiep t a m giac A B C .

la d u o n g p h a n giac ciia goc tao b o i hai d u o n g thang d va d ' .

Jiuang

JCffigidi.

/

01

1) Ta tha'y A va B n a m ve m o t phia so v o i d u o n g thang d. G o i A ' la d i e m d o i

'(x - 2 ) ( - 7 ) + (y - 1 ) ( - 4 ) = 0

J7x + 4y - 1 8 = 0
[Sx + 2y - 26 = 0

(x - 4)(-5) + (y - 3)(-2) = 0 ^

Vay H

13x + 1 4 y - 4 3 = 0

iL

<=> <

5 ,

J_

10

x-2y-3 =0

bo
ok
•M

= (X

- 4)2 + (y - 3)2

( x - 2 ) 2 + ( y - l ) 2 =(x + 3)2+(y + l)2

Vay I

5 '10

rx = l
<=><^
, suy ra d n A = I ( l ; - l )
3x + 4y + l = 0
[ y - - l

V i A la p h a n giac cua goc h g p b a i g i i i a hai d u a n g thang d va d ' nen d va

Lay E(3;0) G d , ta tim d u g c F

F e d ' . Suy ra F I =

14

U

5'5

45

[8x + 4y = - 5

y = -

4 ' 4

2) D u o n g t r o n ngoai tiep t a m giac A B C c6 ban k i n h R = l A =

N e n no p h u o n g t r i n h la:
Bai

(

'3

_16'

.5'"5

2

25^
x+ — +

, d o d o p h u o n g t r i n h d ' : l l x - 2y - 1 3 = 0 .

—

1385

va

p h u o n g t r i n h hai d u a n g t r u n g t u y e n B M : 3x + 4y - 3 = 0 , C N : 3x - l O y - 1 7 = 0 .
T i n h toa do cac d i e m B, C.
Jiuang

la d i e m do'i x i i n g v a i E qua A , ta c6

45^ ^
y -

V2770

8
4y
V
4,
1 . 1 . 2 . T r o n g m a t p h a n g toa do O x y cho t a m giac A B C c6 A(3;2)

I

d ' do'i x u n g n h a u qua A , do do l e d ' .

(2

25
x= —

fx + y = 5

25_45

16 J _

ww
w.

X = -

fa
ce

16

2) Xet he p h u o n g t r i n h

- 2)2 + (y - 1 ) 2

/g
, do do p h u o n g

x-2y-3=0

I A 2 = IB2

Ta
s/

5

up

5'

trinh A ' B :]3x + 1 4 y - 4 3 = 0

Nen M :

46

I A 2 = IC2

•(X

ro

_6

.c
om

5^

46

-

Goi I ( x ; y ) la t a m d u a n g t r o n ngoai tiep t a m giac A B C , ta c6:

23

26

34

y =

ie

x-2y-3=0

V i H la t r u n g d i e m ciia A A ' nen

5

^

34

uO

2x+y-8=0

nT

19

28

X =

hi

V i A ' A 1 d nen A A ' c6 p h u o n g t r i n h : 2x + y - 8 = 0

Suy ra A ' B =

BH.AC = 0

oc

D a n g thuc xay ra k h i va chi k h i M = A ' B n d .

yA' = 2 y H - y A = - 5

AH.BC = 0

Da

iH

Dodo: M A + MB = A ' M + M B > A ' B .

•A'

ddn gidi

1) Goi H ( x ; y ) la true t a m t a m giac A B C , ta c6:

x u n g v o i A qua d. K h i do v a i m o i d i e m M thuoc d, ta l u o n c6: M A = M A '

'23

Viet

CP BAI TAP

m a t p h a n g O x y cho d u o n g thang d : x - 2 y - 3 = 0 va hai

Goi H = d n A A ' = > H : < ^

Khang

dan gidi

:?; • ;

Goi G la t r o n g t a m ciia t a m giac, suy ra toa do ciia G la n g h i e m cua he
'3x + 4y - 3 = 0
3x-10y-17 = 0

7
^ = 3
[y = - l

>

;

r

J..J' . I ' i -


Phumig phdpgiiii

Toan Hitih hoc theo chuyen de- Nguyen Phi'i Khanh, Nguyen Tat Thu

Goi E la t r u n g d i e m ciia BC, suy ra EA = - G A => E(2;

Cty TNHH MTV DWH

.

Jiic&ng ddn gidi
Ta CO p h u o n g t r i n h B C : x + 2y + 5 = 0 .

Gia sir B ( a ; b ) , suy ra C ( 4 - a ; - 5 - b ) . T u do ta c6 h^:
3a + 4b - 3 = 0

3a + 4 b - 3 = 0

" 3(4-a)-10(-5-b)-17 = 0

a=5

[-3a + 10b + 45 = 0

Tpa d p d i e m C la n g h i e m '^"^

b = -3'

/

01

oc

p h u o n g t r i n h hai d u o n g phan giac t r o n g B D : x - y - 1 = 0,CE : x + 2y + 1 7 = 0 .

Toa d p d i e m A la nghiem ciia he:

Gpi A^ d o i x i i n g v o i A qua BD, suy ra A j e BC va A ^ ( l ; - 4 )
A 2 ( - — ; - — ) .

5

ie

x + 2y + 17 = 0

fx = - 3

3x-4y-19 =0

[y = - 7

•C(-3;-7).

cao

d u o n g thang A C .

/g

A A ' : x - y + 2 = 0,

Ta CO p h u o n g t r i n h BC: x + y - 2 = 0
Suy ra toa d o ciia B la n g h i e m cua he:

fa
ce

ddn gidi

tuyen

ddn gidi

|'7x-2y-3 = 0
Toa d o A thoa m a n he: <^
•
•
[6x-y-4 =0

A(l;2)

V i B do'i x i i n g v o i A qua M nen suy ra B = (3; - 2 ) .
D u o n g thSng BC d i qua B va v u o n g goc v o i d u o n g thSng: 6x - y - 4 = 0 nen

P h u o n g t r i n h B C : x + 6y + 9 = 0 .

fx = - l

2x + 5 y - 1 3 = 0

ly = 3

Gpi A ( a ; a + 2 ) , suy ra toa do ciia t r u n g d i e m A C la M

Jiu&ng

suy ra

x+y-2=0

ww
w.

Jiixang

trung

bo
ok

B M : 2x + 5y - 1 3 = 0 . T i n h toa d o cac d i e m A , B.

duong

.c
om

duong

va 6x - y - 4 = 0 . Viet p h u o n g trinh

Ta

[ y = -16

s/

3x-4y-19 =0

l u p t CO p h u o n g t r i n h la 7x - 2y - 3 = 0

B(-15;-16).

B(-15;-16),C(-3;-7).

trinh

2

iL

x = -15

up

x-y-l=0

1 . 1 . 4 . T r o n g m a t phSng toa do O x y cho t a m giac A B C c6 C ( 5 ; - 3 ) va

phuong

5;6 , C ( 9 ; - 7 ) .

la t r u n g d i e m cua canh A B . D u o n g t r u n g t u y e n va d u o n g cao qua d i n h A Ian

ro

Toa d o C la n g h i e m cua he:

2

Bai 1.1.6. T r o n g m a t phang v o i h^ tpa d p O x y , cho t a m giac A B C co M (2; 0)

5

Suy ra p h u o n g t r i n h BC : 3x - 4y - 1 9 = 0 .
Toa d p B la n g h i ^ m cua he:

y =6

nT

va

Vay A

2 => A

uO

do'i x u n g v o i A qua CE, suy ra A 2 e BC

2x + y - l l = 0

Da

ddn gidi

5

2x-y+l=0

hi

Jiu&ng

iH

T i n h toa d o cac d i e m B, C.

Bai

C(9;-7).

D o do, ta CO p h u o n g t r i n h A C :2x + y - l l = 0 .

Bai 1.1.3. T r o n g m a t phang toa d o O x y cho t a m giac A B C c6 A ( - 3 ; 0 ) va

Vay

[x + y - 2 = 0
[x = 9
L ^ 2y + 5 = 0 ^ |y = - 7

Gpi B' la d i e m d o i x u n g v o i B qua CE, suy ra B'(5;l) va B' e A C

Vay B ( 5 ; - 3 ) , C ( - l ; - 2 ) .

Aj

Khang Viet

Tpa d p t r u n g d i e m N cua BC thoa m a n he:
•B(-l;3).

+ 5 a-1^

'7x-2y-3 = 0
'x+6y+9=0

•N

Suy ra A C = 2 . M N = (-4; - 3).
P h u o n g t r i n h d u o n g th^ng A C : 3x - 4y + 5

0.

^ ;

Bai 1.1.7. T r o n g m a t phSng O x y cho d u o n g t r o n ( C ) : (x - if + (y - 1 ) ^ = 25 .
Ma M e B M

nen 2 ^ y ^ + 5 ^ - 1 3

= 0 « a = 3 =^ A ( 3 ; 5 ) .

1) L a p p h u o n g t r i n h tiep tuyen cua (C), biet tiep tuyen d i qua A ( 3 ; - 6 )

Vay A ( 3 ; 5 ) , B ( - 1 ; 3 ) .

Bai 1.1.5. T r o n g m a t phang toa d p O x y cho tam giac A B C

2) T u d i e m D ( - 4 ; 5 ) ve de'n (C) hai tiep tuyen D M , D N ( M , N la tiep diem). Viet
CO B ( l ; —3) va

p h u o n g t r i n h d u o n g thang M N .

p h u o n g t r i n h d u o n g cao A D : 2 x - y + 1 = 0 , d u o n g phan giac C E : x + y - 2=::0
. T i n h toa d p cac d i e m A , C.

,,

J^lurnigd&ngidi
D u o n g t r o n (Qxd-taDL.K2i 1), ban k i n h R = 5 .
T H U

ViEN Tl.VHBtNH

THU.AN]

(.

,ob « . M ( +
Av isH
1


Cty TNHH MTV DWH

Phumig phtip giai Toan Hinh hoc theo chuyen dS"- Nguyen Phu Khdnh, Nguyen Tat Thii

1) Gia six A : ax + by + c = 0 la tiep tuyen ciia (C)

+) Neu diem M e A : ax + by + c = 0,a ^ 0 thi M

Do B e A nen 3a - 6b + c = 0 => c = 6b - 3a
2a + b + c

A la tiep tuyen ciia (C) nen d(I, A) = R

Va^+b^

= 5<=>

-a + 7b

-bm-c

Khang Viet

- ; m , liic nay toa

do ciia M chi con mgt an va ta chi can tim mgt phuong trinh.

=5

Vi da 1.2A. Trong mat phang Oxy cho duong tron (C): (x - 1 ) ^ + (y - 1 ) ^ = 4
va duong thang A : x - 3 y - 6 = 0. Tim tga dg diem M nam tren A , sao cho
tvr M ve dugc hai tiep tuyen M A , MB (A,B la tiep diem) thoa AABM la tam
giac vuong.

oc

a=-ib
3

Duong tron (C) co tam 1(1; 1), ban kinh R = 2 .

Da

3x + 4y +15 = 0 va 4x - 3y - 30 = 0 .

hi

Vi AAMB vuong va I M la duong

fTe(C)

nT

phan giac ciia goc AM B nen A M I = 45°

DI.IT = 0

uO

Trong tam giac vuong l A M , ta co:

ie

(Xo-2)2+(y„-l)2=25

I M = 2V2, suy ra M thugc duong
tron tam I ban kinh R' = 2

Ta

Xo+yo-4xo-2yo=20

iL

(xo-4)(xo-2) + (yo4-5)(yo-l) = 0
_
,
^
^2xo-6yo-23 = 0

s/

<=>

.c
om

§ 2. X A C D I N H T O A D O C U A M Q T D I E M

/g

ro

up

Xo+yo-6xo+4yo=-3
Vay phuong trinh M N : 2x - 6y - 23 = 0 .

bo
ok

Bai toan co ban ciia phuong phap toa do trong mat phang la bai toan xac

dinh toa do ciia mot diem. ChSng han, de lap phuong trinh duong thang can

fa
ce

tim mot diem di qua va VTPT, voi phuong trinh duong tron thi ta can xac djnh
tarn va ban kinh....Chung ta co the gap bai toan tim toa do ciia diem dugc hoi

ww
w.

true tiep hoac gian tiep.

Xgigiai

iH

T u do, ta CO dugc phuong trinh tiep tuyen la:

2) Goi T ( X ( , ; y Q ) la tiep diem , ta c6:

01

/

4

o l 2 a ^ +7ab-12b^ = 0 «

.

Mat khac M e A nen M

la giao

diem ciia A va ( I , R ' ) . Suy ra tga do
ciia M la nghiem ciia he
x-3y-6=0

x=3y+6

( x - i ) 2 + ( y - i ) 2 =8 "
'x = 3y + 6
5y^ +14y + 9 = 0

[(3y + 5)2 + ( y - l ) ' =8

y= -l,X= 3

= -^~

=-

5'^

5

(3
9I
Vay CO hai diem M j (3; - l ) va M 2 - ; — thoa yeu cau bai toan.

• Ve phuong dien hinh hgc tong hgp thi de xac dinh toa do mot diem, ta

Vi du 1.2.2. Trong mat phSng voi he tga do Oxy cho cac duong thang

thuong chiing minh diem do thugc hai hinh (H) va (H'). Khi do diem can tim

d i : x + y + 3 = 0, d j : x - y - 4 = 0, dg : x - 2 y = 0. Tim tga do diem M nam

chinh la giao diem ciia (H) va (H').

tren duong thSng

• Ve phuong di^n dai so, de xac dinh toa do ciia mot diem (gom hai toa do) la

hai Ian khoang each t u M den duong thang d2 .

bai toan di tim hai an. Do do, chiing ta can xac djnh dugc hai phuong trinh
chiia hai an va giai he phuong trinh nay ta tim dugc toa do diem can tim. Khi
thiet lap phuong trinh chiing ta can luu y:
+) Tich v6 huong ciia hai vec to cho ta mgt phuong trinh,
+) Hai doan thang bang nhau cho ta mgt phuong trinh,
+) Hai vec to bang nhau cho ta hai phuong trinh,
18

'

sao cho khoang each t u M den duong thang d^ bang

Xffi gidi

-4
3y + 3
Taco M e d 3 , s u y r a M ( 2 y ; y ) . Suy ra d ( M , d i ) = — ^ ; d ( M , d 2 ) = ^ ^
Theo gia thiet ta co: d ( M , d i ) = 2d(M,d2) <^

3y + 3 ^ 2 l y - 4


Cty TNHH
Phuvng

plidp gidi Toiin Hiith hoc theo chuyen

dc- Nguyen Phii Klidnh,

Nguyen

MTV

DWH

Khang

Viet

Tn't Thu

Vi du 1.2.5. Cho parabol (P): y^ = x va hai d i e m A(9; 3), B ( l ; -1) thupc (P).
3y+3=2y-8

<=>

3y + 3 = - 2 y +

G p i M la d i e m thupc c u n g A B cua (P) (phan ciia (P) bi chan b o i day

<=> y = - l l ; y = 1 .

Xac d j n h tpa d p d i e m M n a m tren cung A B sao cho t a m giac M A B c6 dien
tich i o n nha't.

• Voi y = - n ^ M ( - 2 2 ; - l l ) .

JCgi gidi.

• Voi y = l ^ M ( 2 ; l ) .
Phuong trinh

O x y , cho d i e m A(0; 2) va d u o n g th3ng

V i M G (P) => M ( t ^ ; t) t u gia thiet suy ra - 1 < t < 3

01

d : x - 2 y + 2 = 0 . T i m tren d u o n g thang d hai d i e m B, C sao cho t a m giac

T a m giac M A B c6 dien tich i o n nha't o

oc

A B C v u o n g 6 B va A B = 2BC .

Ma d(M;AB) =

'2.6'

x-2y+2=0
TQa d p d i e m B la n g h i e m ciia he : <| ^
^ \

Suy ra m a x d ( M , A B ) =

nT

5'5

ie

Vi du .2.6. T r o n g m a t p h a n g Oxy cho d u o n g

—

5,

x-2y + 2= 0

I

—

5,

f

+

v

y "

5

5y

x=ay=l
4

6l
5

5j

Vay CO hai bp d i e m thoa yeu cau bai toan la:

7

fa
ce

bo

ok

' 2 6^
4 7
2 6
5 ' 5 , C ( 0 ; l ) va B 5 ' 5 , C

Vi du 1.2.4. T r o n g m a t p h a n g v o i h ^ tpa dp O x y , cho d i e m A(2; 2) va hai

ww
w.

d u o n g thSng: d i : ) ^ + y - 2 = ^ 0 , d 2 : x + y - 8 = 0 . T i m tpa dp d i e m B, C Ian
l u p t thupc d i , d2sao cho tam giac A B C v u o n g tai A.
Xffigidi

AB = AC
Dat x = b - l ; y = c - 4 t a c o :

Ta CO A B = Vio

va S^^^AB = - d ( M , A B ) . A B =

x =2

<=> •

.y

= i

d(M,AB) =

Lai CO A B = (1;3) nen n = ( 3 ; - l ) la VTPT ciia d u o n g t h a n g A B
Suy ra p h u o n g t r i n h A B : 3(x - 1 ) - ( y +1) = 0 hay 3 x - y - 4 = 0 .
Gpi M ( a ; b) e (C) => (a - i f + b^ = 2
,
K h i do d ( M ; A B ) - - =

<=>

VIO

3a - b - 4 = 1

3a-b-4
; = — -

vio

^ 3a-b-4

=1

VlO

f(a-l)2 + b2=2 , .
<=i>r
hoac
3a-b-4 = l

( a - l ) 2 + b^ = 2

(b-lf-(c-4f

= 3

Xffi gidi.

(a-l)2+b2=2

f(b-l)(c-4) = 2

xy = 2
. x ^ - y ^

dien tich tam giac M A B bang ^ .

Ta CO he p h u o n g t r i n h :

B6di=^B(b;2-b);Ced2=^C(c;c-8).

Theo de bai ta c6 he:

iL

1

-—

ro

2^

X

V

—

/g

(

y -

.c
om

Vay toa d p d i e m C la n g h i e m ciia he :

6^

+

B(2;2). T i m tpa d i e m M thupc d u o n g t r o n (C) sao cho

Ta

I

2^

X

tron ( C ) : (x - 1 ) ^ + y^ = 2 va

s/

la: (

hai d i e m A ( l ; - 1 ) ,

2

up

2

P h u o n g t r i n h d u o n g t r o n tarn B, ban k i n h BC = —
5

Xi.AC = 0

dat d u p e k h i t = 11=> M ( l ; l ) .
v5

uO

S

„
2^5
^„
AB
Suy ra A B =
=> BC = - — = — .
^
5
2
5

Vi

t^ - 2 t - 3
-,te(-];3).

hi

Ta CO A B 1 d nen A B c6 p h u o n g t r i n h : 2x + y - 2 = 0 .

d ( M , AB) Ion nha't

iH

JCffigidi

B

A B : x - 2y - 3 = 0

/

1.2.3. T r o n g he toa do

Da

Vi du

AB).

.3

b = 3a - 5

x - -2
V <
.y

=

-1

(a-1)2+(3a-5)2
b = 3a - 5

hoac
=2

(a-l)2+b2=:2

i)k>,J.

3a-b-4 = - l

( a - l ) 2 + b2 = 2
b = 3a - 3
hoac

(a-1)'+(3a-3)2 =2

.vi ,

h . ; ^ / , „

b = 3a-3

Vay B ( 3 ; - 1 ) ; C ( 5 ; 3 ) hoac B ( - 1 ; 3 ) , C ( 3 ; 5 ) .
21


Phuang phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

5a^ - 1 6 a + 12 = 0

hoac

b = 3a-5
_12

_4

<=>^ "i T ' ' ' " 5 hoac
b = 3a-5

Cty TNHH MTV DWH Khang Viet

5a^ -10a+ 4 = 0
b = 3a-3

I M = 4IN <::>

(2n-l)2 =n2

V

m =2

2 •
m =—
3

va M 4

/

Bai 1.2.3. Trong mat phang toa dp Oxy cho diem A(3;2), cac duong thang

5 + V5 375

01

13^ _ f S - V S -375

dj : X + y - 3 = 0 va: d2 : x + y - 9 = 0 . Tim toa do diem B G d j , va C e d2 sao

5

iH

m BAI TAP

Jiuongr , s ' r .
Vi B e d j :x + y - 3 = 0 nen B ( b ; 3 - b ) , C e d j :x + y - 9 = 0 nen C ( c ; 9 - c ) .

hi

nT
uO

phuong trinh: (x -1)^ + y^ = 1. Ggi I la tarn ciia (C). Xac dinh toa do diem M

De tam giac ABC vuong can tai A khi va chi khi

Ta

+h^=l.

up

Mat khac O € (C) =:> lO = I M = 1.

/g
3
a=—
2

bo
ok

(a-l)'+b2=l

ww
w.

Til

Bai 1.2.2. Trong mat ph^ng voi he toa do Decac vuong goc Oxy, cho

u ^ + ( u + 2)^ = ( v - l ) 2 + ( v - 5 ) ^

Dat u = b - 3, v = c - 2, ta c6:

u ( v - l ) + (u + 2 ) ( v - 5 ) = 0
(u + 1)^ = ( v - 3 ) ^ + 3
uv-3u+v-5=0

b=+

fa
ce

a^+b^ = 3

2

.c
om

nen OM^ = lO^ + I M ^ - 2IO.IM.cos 120° o a^ + b^ = 3.

AB.AC =0

(b-3)(c-3) + (l-b)(7-c) = 0

ro

Tam giac I M O c6 O I M - 1 2 0 °

Toa do diem M la nghiem cua he :

Hay

s/

Ggi diem M ( a ; b ) . D o M e ( C ) nen {a-lf

AB2=AC2

<=> i

(b-3)2+(b-l)2=(c-3)2+(c-7)2

iL

Jiudng ddn gidi

AB =AC
AB 1 AC

ie

thuQC (C) sao cho I M O = 30".

^

,

Da

cho tam giac ABC vuong can tai A.

Bai 1.2.1. Trong mat phang voi he toa do Oxy, cho duong tron (C) c6

Vay M =

<=> s

1

n =—
3
{

oc

5

r4

m - 2 = 4(n - 2)

n=l

Vay CO hai cap diem thoa yeu cau bai toan la:
1l ^
M(4;2),N(1;1) hoac M
9' 3J
9'3

b = 3a-3

12 m

m =4n-2

5±V5
a=•

Vay CO boh diem thoa dieu kien bai toan la:

Ml

m^ = 4 n 2

3u + 5

•+3

,

v=
(u +

<=>

3u + 5
v=•
u +1

u = -3

u=l
u +1
<=><^
x2 = 4
V =
1)^

.

I

V •!

4

v=2

Vay CO hai cap diem thoa yeu cau bai toan la:

,.j *

B(4;-1),C(6;3) hoac B(0;3),C(4;5).

parabol (P) c6 phuong trinh y^ = x va diem 1(0; 2). Tim toa do hai diem M , N

Chti i/.-Ngoai each tren, ta c6 the giai theo each khac nhu sau:

thuQC (P) sao cho I M = 4iN .

Tjnh tien he true toa dp Oxy ve he tuc XAY theo vec to O A , ta c6 cong thuc
Jiuong dan gidi

Vi M , N e ( P ) nentaco M ( m ^ ; m ) , N ( n ^ n ) .
S u y r a I M = m ' ^ ; m - 2 , I N = n ^ n - 2 .Dodo:

doi true:

'x = X + 3

'

J =Y +2

=

Trong he true moi, ta c6 phuong trinh cua dj :X + Y + 2 = 0, d2 :X + Y - 4 = 0.
Q : B -> C

Vi tam giac ABC vuong can tai A nen phep quay Q

(A,±90 )

22

;

,

,

,

|,

'23


Cty TNHH MTV DWH

Phumtg phapgiiii Toan Hinh hqc theo chuyen rfe - Nguyen Pht'i Khanh, Nguyen Tat Thu

Bai 1.2.5. T r o n g mat phSng v o i he true toa do Oxy, cho d u o n g thSng

=> C e d 1 = Q^^.^^^jo/cli), do do C ^ d2 n d , .

j . x - 3 y - 4 = 0 va d u o n g t r o n ( C ) : x^ + y^ - 4y = 0 . T i m M thugc d va N

• Xet phep quay Q^^ ^^^^^, ta c6 p h u o n g t r i n h d , : X - Y - 2 = 0
X-Y-2=0

Do do toa d p cua C la nghiem cua he:

<=> {

X+Y-4=0

thupc (C) sao cho c h u n g d o i x u n g qua A ( 3 ; l ) .

X = 3

x=6

Y=r

y = 3

Jiuang
Vi M e d

ddn

1,,;f.„f;

gidi

M ( 3 m + 4; m ) . D o N d o i x u n g v o i M qua A nen N(2 - 3 m ; 2 - m)

MaNe(G)
nen (2 - 3m)^ + (2 - m)^ - 4(2 - m ) = 0 o lOm^ - 12m - 0 o

m = 0,m = -

oc

D o do tga do cua C la nghiem cua he: • ^ ^ ^ ^ ^ <=>
Y = 3'
^ •
• X+Y-4=0

01

x=4

X =l

T.r..^

/

Xet phep quay Q^^ ^^^(y ta c6 p h u o n g trinh d j : X - Y + 2 = 0

y = 5-

Vay CO hai cap d i e m thoa yeu cau bai toan:

T u do ta t i m d u g c B, C.

Da

M(4;0),N(2;2)

iH

Ma B e

Khang Vie I

va M

386

'

8 4^
•5'5

I 5 '5j

hi

Bai 1.2.4. T r o n g he true toa do O x y cho AABC v o i A(2;3), B(2;l), C ( 6 ; 3 ) ,

Bai 1.2.6. Trong mat phSng Oxy cho diem A ( l ; 4 ) . T i m hai d i e m M , N Ian l u g t

d i e m M thuoc d u o n g t r o n ( G ) : (x - 3)^ + (y -1)^ = 25 sao cho : 5^^^ = 2SADB •

nam tren hai d u o n g t r o n ( q ) : ( x - 2 ) 2 + ( y - 5 ) 2 =13 va {C^):{x-lf+{y-2f

uO

sao cho t a m giac M A N v u o n g can tai A .

gidi

ie

ddn

2

BG

3

3

^3

I

3^

3

3^

up

AG

= 1-2.^ = 1

.c
om

nen d ( D , AB) = | =^ S^^BD = {AB.d(D,AB)

/g

ro

P h u o n g t r i n h A B : x - 2 = 0,

P h u o n g t r i n h DG : x - 2y = 0 . Goi M(a; b) => (a - 3)^ + (b -1)^ = 25
a-2b

fa
ce

-GD.d(M,GD) = - « i
ix/s.
2
^ 3 2 3

bo
ok

M a t khac:
' A M C D = 2S AABD

s/

Ta

Ta CO A B = (0; 2), A C = (4; 0), BG - (4; 2)
DG

8
= — <=>a - 2 b
3

=4

a = 2b - 4 thay vao (1) ta c6 duoc: (2b - 7)^ + (b -1)^ = 25 <=> b^ - 6b + 5 = 0

a = 2b + 4 thay vao (1) ta c6 dugc: (2b +1)^ + (b -1)^ = 25 <r> 5b^ + 2b - 23 = 0
=e>a =

^—=>M

.
-1-2729
18-4729
b =:
=> a =

24

.M

- 1 + 2729 18 + 4729
5 .
'-1-2729

Ma

'

5

N va ( G j ) ^ ( G j )

• V o i Q^^ ^ ^ ( y ta CO p h u o n g t r i n h ( G j ) : x ^ + ( y - 5 ) ^

x2+(y-5)^=13

x2 + y 2 _ i 0 y + 12 = 0

(x-i)2+(y_2)2=25

x2 + y 2 - 2 x - 4 y - 2 0 = 0

-13

x 2 + y 2 - 1 0 y + 1 2 = : 0 ^ J5y2 - 53y +134 = 0
x = 3y-16

x = 3y-16
X -

-1-37T29

-1 + 37T29
10

X =V <

10

53-7129
y =10

T r u o n g h o p nay c6 hai bp d i e m :
M

y

18-4729^

gidi

Me(G,)^N€(Gj)^N€(G2)n(Gi).

53 + 7129
y =10

b = 5=>a = 6=>M(6;5)

b =

Xet phep quay Q^^.^^^,0) : M

ddn

Toa d p d i e m N la n g h i e m cua he:

(1)

ww
w.

o a = 2b + 4 hoac a = 2b - 4

"b = l = ^ a = - 2 = > M ( - 2 ; l )

Jiuung

iL

Jiuang

nT

Goi D la giao d i e m cua d u o n g phan giac trong goc B A G v o i BG. T i m tat ca cac

Va M

23 + 7T29 _ 51 - 37T29
10
'23 - 7129
10

'

f - l + 37l29 53 + 7l29
10

10

10

51 + 37l29

-l-37l29

53-7l29

10

10

10

^75


Cty TNHH MTV DWH Khang Vi?t

Phumtg phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phti Khihih, Nguyen Tat Thu

Jiuang ddn gidi

Q(A-9oO)'*^™P''^""^*""^ ( C i ) : ( x - 2 ) 2 + ( y - 3 ) 2 =13

Toa do diem N la nghiem ciia he:

( x - 2 ) 2 + ( y - 3 ) 2 =13
( x - ] ) 2 + ( y - 2 ) 2 =25

{x = A

<=> <

ly = 6

\x = 5

V <

T m o n g h g p nay c6 hai bo diem: M(-1;7),N(4;6) va M(0;8),N(5;5).

Khi do di?n tich tam giac A B C la:

[y = 5

I

01

/

va duong thang d : 5x + 2y -11 = 0. Tim diem C tren d sao cho tam giac ABC
CO trong tam G nam tren duong tron (C) biet A(l;2),B(3;-2).
Jiu&ng ddn gidi
Ta c6: C e d nen ta c6 toa do C c;-11-5C
Tpa do trong tam G c + 4 l l - 5 c . Do G nam tren duong tron (C) nen ta c6

oc

2

iH

ie

uO

nT

hi

Da

hai diem A(3;-2), B(-3;2). Tim tren (E) diem C c6 hoanh dp va tung dp
duong sao cho tam giac A B C c6 dien tich Ion nha't.
>ji
Jiuang ddn gidi
<
Ta CO phuong trinh duong thang A B : 2x + 3y = 0
,

ABC

la

SABC

(X(,

XQ

+

=^

dien tich tam giac

= 2^^-^(C'^^) = ^ | 2 ^ +M = 3 ^ ^ 3 4

ro

up

s/

Ta

iL

Gpi C (x; y) voi x > 0, y > 0. Khi do ta c6

2

'85
< 3 13'

/g

XQ

.2

'
X
Bai 1.2.10. Trong mat phang
voi he toa dp Oxy cho elip (E): —
+ —y = 1 va

.c

om

bo
ok

fa
ce

^ = iS
IM^ = 20
sin30"
Do M e d nen suy ra M ( X Q ; + 1 )
Khi do ta c6: MI^ = + 1 ) ^ + (x„ -1)^ = 20 o x^ = 9 x^ = 3; = -3
Vay CO 2 diem M thoa man dieu kien bai toan: (3; 4); (-3;-2)
Bai 1.2.9. Trong mat phang voi he toa dp Oxy cho diem C(2;-5) va duong
th^ng A : 3x - 4y + 4 = 0 .Tim tren A hai diem A va B doi xung nhau qua 1(2; | )
sao cho dien tich tam giac A B C banglS.
26

ww
w.

Do do: MI =

A) = 3AB

"a = 4
r6-3a^ 2
2 , = 25c* a = 0
Vay hai diem can tim la A(0;1) va B(4;4).

Theo gia thiet ta c6: A B = 5 <^ (4 - 2a)^ +

Bai 1.2.7. Trong mat phSng Oxy cho duong tron (C): (x - if + (y - 4)^ = y

phuong trinh: i ^ ^ + i ^ ^ l H L = ^ <:>29c2 + 114c4-85 = 0 <^ c =-l,c =
9
36
9
29
85.372
Vay CO hai diem C thoa yeu cau bai toan la: Cj (-1;8), C29' 29
Bai 1.2.8. Trong he toa dp Oxy cho duong thang d : x - y + l = 0 va duong
tron (C) CO phuong trinh x'^ + y^ + 2x - 4y = 0. Tim diem M thuoc duong thSng
d sao cho tir M ke dupe hai duong thing tiep xuc vai duong tron tai A va B,
sao cho AMB = 60" .
Jlucrng ddn gidi
Duong tron c6 tam I(-l;2) va ban kinh:R = Vs .
Tam giac AMB la tam giac deu va MI la phan giac goc AMB nen IMA = 30°

S^BC = ^ A B . d ( C ,

Dau bang xay ra khi 9

4

x = 3 2 . Vay C

9
(

2^

1170
=
3
4
13

3 V 2

3 2
§ 3. N H O M C A C B A I T O A N V E

HlNH

BINH

HANH

Khi giai cac bai toan ve hinh binh hanh, hinh thoi, hinh chu nhat va hinh
vuong, chung ta can chu y den tinh chat doi xung. Chang han, giao diem hai
duong cheo la tam doi xiing cua hinh binh hanh; hai duong cheo ciia hinh thoi
lajrycdoi xung....
_
Vi du 1.3.1. Trong mat phSng Oxy cho hai duong thang di: x - 2y + 1 = 0,
d2: 2x + 3y = 0. Xac djnh tpa dp cac dinh cua hinh vuong ABCD, biet A thupc
j y o n g thang di, C thupc duong thang d2 va hai diem B, D thupc true Ox.
27

Phucnig plidp giai Toan Hinh hgc theo chuyen

de - Nguyen

Phu Kluhili,

Nguyen

Tat

Thu
Cty TNHH

Xgi gidi.
Vi A e d , , C e d 2

nen A(2a - l ; a ) , C ( 3 c ; - 2 c ) , suy ra I

r2a + 3 c - l

a-2c

a(x + 3) + b(y - 6) =

la

D o A B C D la h i n h v u o n g nen 1 la t r u n g d i e m cua BD, hay I e Ox .

nen ta co:

DWH

Khang

Viet

0 o ax + by + 3a - 6b = 0

V i d ( M , E Q ) = 7i0

trung diem A C

MTV

^

5a-5b

E

= >/To

ci>(5a-5b)^ =10(a2+b2)

c = 1.

01

M a t khac A C 1 BD = Ox nen suy ra 2a - 1 = 3c o

<:i>3a^ - 1 0 a b + 3b^ = 0 <=> a = 3b,b = 3a
,

b = 5,b = 1 .

a = 3 b , ta c6 p h u o n g t r i n h E Q : 3x + y + 3 = 0 .

iH

V i B e Ox =^ B ( b ; 0 ) , ma IB = I A = 2 =^ |b - 3| = 2 o

oc

T u do, ta t i m d u g c A(3;2), C(3;-2), 1(3; 0 ) .

Da

V a y toa d o cac d i n h ciia h i n h v u o n g A B C D la:
A(3;2), B(1;0), C ( 3 ; - 2 ) , D(5;0) hoac A(3;2), B(5;0), C ( 3 ; - 2 ) , D ( 1 ; 0 ) .

hi

K h i do toa d o Q la nghiem ciia he

7.3.2.Trong m a t phang O x y cho ba d i e m 1(1; 1), J(-2;2), K ( 2 ; - 2 ) . T i m

s/
up
ro

/g

=> I A = 4

I A = 4 o ( a - l ) ^ +(a + 3)^ = 1 6 « . a ^ + 2a-3

= 0<=>a = l , a = - 3

a = 1 , ta C O A ( l ; 3 ) , B ( - 3 ; l ) , C ( l ; - 1 ) ,

•

a = - 3 , ta C O A ( - 3 ; l ) , B ( l ; 3 ) , C ( 5 ; l ) , D ( l ; - 1 ) .

ww
w.

•

Vidu

D(5;l)

fa
ce

A e A B => A ( a ; 4 + a ) , do do

2.3.3.Trong mat phang O x y cho d u o n g tron (C): (x - 2)^ + (y - 1)^ = 10.

y = 0

uO

ie
iL

4V2

nen suy ra

bo
ok

AB =

.c
om

AB:x-y+4=0.
D o d ( I , A B ) = 272

he

Ta

Ciia

V i A B / / C D nen p h u o n g t r i n h

3x + y + 3 = 0

• b = 3a, ta CO p h u o n g t r i n h E Q : x + 3y - 1 5 = 0 . K h i do toa d o Q la nghiem

canh A B va K thuoc canh C D .

Ta c6: KJ' = (2; 2), suy ra p h u o n g t r i n h C D : x - y - 4 = 0.

x =- l

T r u o n g h o p nay ta loai v i X Q > 0 .

toa d o cac d i n h cua h i n h v u o n g A B C D sao cho I la tam h i n h v u o n g , J thuoc

G o i J' d o i x i i n g v a i J qua I , ta c6 J'(4;0) va J' € C D .

(x-2)2+(y-l)2=10

nT

Vidu

/

D o do a = 2c .

(x-2)2+(y-l)2=10
3x + y + 3 = 0

<=>

fx = 3
<

•Q(3;4).

y = 4

T a c o P ( 1 5 - 3 x ; x ) va QP = M Q => (12 - 3x)^ + (4 - x f = 10

x = 3,x = 5

X = 3, ta CO P ( 6 ; 3 ) , suy ra t a m cua h i n h v u o n g 1(4;2) nen N ( 5 ; 0 )
X = 5, ta CO P ( 0 ; 5 ) , suy ra tam cua h i n h v u o n g 1(1;3) nen N ( - l ; 2 ) .
Vay CO hai bo d i e m thoa yeu cau bai toan:
M(2;1),N(5;0),P(6;3),Q(3;4) va M ( 2 ; 1 ) , N ( - 1 ; 2 ) , P ( 0 ; 5 ) , Q ( 3 ; 4 ) .
Vi du 1.3.4. T r o n g mat phang v o i he toa do O x y cho h i n h chir nhat

CO d i e m 1(6; 2) la giao d i e m cua 2 d u o n g cheo A C va B D . D i e m M ( 1 ; 5)
thuQc d u o n g th3ng A B va t r u n g d i e m E ciia canh C D thuoc d u o n g thang
d : x + y - 5 = 0 . Viet p h u o n g t r i n h d u o n g thang A B .
gidi.
Vi E€d=:>E(a;5-a)=>iE = ( a - 6 ; 3 - a ) .

T i m toa d o cac d i n h cua h i n h v u o n g M N P Q , biet M t r u n g v o i tam cua

Goi N la t r u n g d i e m cua A B , suy ra I la t r u n g d i e m cua E N nen :

d u o n g t r o n (C); hai d i n h N , Q thuoc d u o n g t r o n (C); d u o n g thang PQ d i qua

N:

E ( - 3 ; 6 ) va X Q > 0 .

X^

=2xi - X g

XN = 2 y , - y E

=12-a
=a-l

•N(12-a;a-l)

•MN-(ll-a;a-6).
Ta C O M ( 2 ; l ) va EQ la tiep tuyen cua ( C ) .
P h u o n g t r i n h EQ c6 dang:
28

ABCD

V i E 1 M N => M N . I E = 0
< » ( l l - a ) ( a - 6 ) + ( a - 6 ) ( 3 - a ) = 0<r>

a= 6
a = 7'


gidi Todn Hinh hoc theo chuyen

de-

Nguyen

Phu Khdnh,

Nguyen

Tat

Cty

Thu
'

a = 6 => M N = (5;0), suy ra phuang trinh

Ta

AB:y-5 = 0
•

co:

C

_

b^^^^

- -^ABCD - y

15 _ 5

A T -

-

-

1
Ma A e d g => A(a; a + 2) => A r =2 a —
2

a = 7 => M N = (4;1), suy ra phuong trinh A B : x - 4y +19 = 0.

Vi du 1.3.5. Trong mat phang voi he true toa do Oxy cho hinh chu nhat
ABCD CO dien tich bang U, tam I la giao diem cua duong thang
d i : x - y - 3 = 0 va d 2 : x + y - 6 = 0. Trung diem cua AB la giao diem aia

nen ta c6: a

1

25

TNHH

MTV

DWH

Kha„g

Viet

.AI^=^
^

\

''no

o a = 3,a = -2

/

Phumtgphdp

01

2
Vay toa do cac dinh ciia hinh thoi la:

oc

dj voi true Ox. Tim toa dp cac dinh cua hinh chi> nhat.

iH

A(3;5),B(2;1),C(-2;0),D(-1;4) hoac A(-2;0),B(2;1),C(3;5),D(-1;4).

x + y-6 =0

yi

1,

Vidu. 1.3.7. Trong mat phMng he toa do Oxy, cho hinh thoi ABCD c6 tam

Da

3^

uO

ie

iL

2 7 2 =^ A M = 2

Ta

=

s/

AD

up

Ma A € AB=^ A ( a ; 3 - a ) i = > A M ^ = 2 o ( a - 3 ) ^ = 1 <:i>a = 2,a=4

/g

ro

Tachon A(2;1),B(4;-1).

.c
om

Do I la tam ciia hinh chii nhat nen C(7;2), D(5;4).

bo
ok

Vay toa do cac dinh cua hinh chu nhat la: A(2;l), B(4;-l), C(7;2), D(5;4).
Vi da 1.3.6. Trong mat phang Oxy cho ba duong thang d j : 4x + y - 9 = 0,
d 2 : 2 x - y + 6 = 0,

d 3 : x - y + 2 = 0. Tim toa dp cac dinh ciia hinh thoi

fa
ce

ABCD, biet hinh thoi ABCD c6 dien tich bang 15, cac dinh A, C thuoc ds, B

ww
w.

thuoc di va D thuoc d2.

B = BD n d i , suy ra B
^
^
[4x + y - 9 = 0
Tuong t u D = B D n d j => D

•B

^9-m

2

phuong trinh A B : 4x + 3y - 1 = 0 .

M

c

Vi AC = 2BD nen A I = 2 B I .
Goi H la hinh chieu ciia I len AB, ta c6:
IH = d(I,AB) =

8 + 3-1

= 2 va

1
IH^

B
1

lA^

1
-+•

IB^

4IB2

.IB =
4b + 2

IHVS

=5ob =l

1

2m-l

Xffi gidi

.2

2

x+y-l =0
fx = - l
Tpa dp giao diem ciia d j va d j la nghi^m ciia h?: ^
<=>
3 x - y + 5 = o'^|y = 2

1
2m-l
+ 2 = 0 < » m = 3.Suyra
Vi IeAC=i> —
2

nen

du. 1.3.8. Trong mat phang Oxy cho hai duong thang di: c + y - 1 = 0,
d2: 3x - y + 5 = 0. Tim toa dp cac dinh ciia hinh binh hanh ABCD, bie't 1(3; 3)
la giao diem ciia hai duong cheo; hai canh ciia hinh binh hanh nam tren hai
duong thang d i , d2 va giao diem ciia hai duong thang do la mpt dinh ciia
J y n h binh hanh.

4m-9^

^ m - 6 2m + 6^

Suy ra tga dg trung diem ciia BD la I

Suy ra M N ' =

Mat khac B e AB => B(b;^—^),b > 0 =^ IB^ = (b - 2 ) ^ +
3
Vay B ( l ; - 1 ) .

V i B D 1 AC nen phuong trinh BD: y = - x + m
y = -x + m

thuoc duong thang A B ; diem N(0;7)

thuoc duong thang CD . Tim toa doXffi
dinh
gidi.B biet B c6 hoanh do duong.
Goi N ' la diem doi xung ciia N qua tam I
D
thitaco N'(4;-5) va N ' thuoc canh A B .

Vi AB 1 M I nen suy ra phuong trinh A B : x + y - 3 = 0
c

l\

I(2;l) va AC = 2BD. Diem M 0 ; -

Goi M la giao cua duong thang dj voi Ox, suy ra M(3;0).

AD = 2MI = 3 V2 =^ AB =

f

hi

(9

nT

Taco dj n d j = 1 :

x-y-3=0

B(2;1),D(-1;4),I

^ 1 5^
2'2


Cty TNHH MTV DWH

Phumig pitdp giai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu
Do do M P = N Q p

D

song song voi A B , suy ra phuang

<-> E P M = F Q N <=> Q I M =

trinh d : x + y - 6 = 0.

2' 2

, suy ra B

1 ^

goc voi MP

la trung diem ciia A D

Suy ra phuong trinh d: x - 4 = 0.

oc

4' 4

Gpi E la giao diem cua d voi duong thang

iH

3 19

.M

7^

AD, ap dung tinh chat tren ta suy ra NE = MP

Da

Do do D

y=

4
23

Gpi d la duong thang di qua N va vuong

hi

3x-y + 5 = 0

X = —

<=>

Taco: MP = (0;-1)=>MP = 1 .

B

Ma E(4;m) nen NE = MP o (m -2)^ = 1 <=> m = 3,m = 1 .

nT

x+y-6=0

1

A

Vidi^ 7.3.9. Cho hinh binh hanh ABCD c6 B(l;5), duang cao AH:x+2y-2=0,

ie

A D : x - 3y + 5 = 0

do cac dinh con lai ciia hinh binh hanh.

iL

Phuong trinh A B : 3x + y - 7 = 0, BC : x - 3y -10 = 0, C D : 3x + y - 6 = 0.

Ta

Xgigidi.

s/

Goi d : X - y - 1 = 0.

ro

up

Phuong trinh BC : 2x - y + 3 = 0 ,

[y = -5^

•C(-4;-5).

.c
om

x-y-l=0

/g

suy ra toa do cua diem C la nghiem ciia he
x = -4

QE = (3; 1), suy ra phuang trinh

uO

• Voi m = 3, suy ra E(4; 3)

duong phan giac trong ciia goc ACB c6 phuang trinh x - y - 1 = 0. Tim toa

2x-y+3=0

90" o M P 1 N Q

Tra lai bai toan:

d

Toa do giao diem ciia d va A D :

AMEP = ANFQ

/

Goi d la dtrong thang di qua I va

Hai tarn giac vuong M E P va N F Q c6 N F = M E .

^

01

Ta gia sir A ( - l ; 2) va AB = d p A D = d 2 , suy ra C(7; 4).

Khang Vi$t

H

C

• Voi m = 1 , suy ra E(4;l)

QE = ( 3 ; - l ) , suy ra phuang trinh

.

AD:x + 3y-7 = 0

^^

Phuang trinh A B : 3x - y - 5 = 0, BC: x + 3y + 2 = 0, C D : 3x - y - 6 = 0.

m BAI

TAP

Bai 1.3.1. Trong mat phang voi h? toa dp Oxy cho hai duang thSng di: x - y = 0,
d2: 2x + y - 1 = 0. Tim tpa dp cac dinh hinh vuong ABCD biet rang dinh A

Suy ra phuang trinh A C : x - 2y - 6 = 0.

thupc d j , d i n h C thuoc d j va cac dinh B,D thupc tryc hoanh.

Vi A D = B C = > D ( - 1 ; - 1 1 ) .

fa
ce

x-2y-6-0

x=4

x+2y-2=0

J^Iu&ng ddn gidi

A(4;-l).

ww
w.

Toa dp diem A la nghiem ciia h$:

bo
ok

Ggi B' do'i xung voi B qua d, ta tim dugc B'{6;0) va B' e AC .

Vidu 7.3.iO.Trong mat phang voi he toa do Oxy cho hinh vuong A B C D biet
M ( 2 ; 1 ) , N ( 4 ; - 2 ) ; P(2;0); Q(1;2) Ian lupt thupc canh A B , B C , C D , A D . Hay

Vi A e d j => A ( t ; t ) , A va C doi xung nhau qua BD va B,D€Ox=>C(t;-t).
Vi C G d 2 r ^ 2 t - t - l = 0 o t = l . V a y A ( l ; l ) , C ( l ; - l ) .
Trung diem cua AC la l ( l ; 0 ) . Vi I la tarn cua hinh vuong nen

l^p phuang trinh cac canh cua hinh vuong.
B , D e Ox ^ B ( b ; 0 ) , D ( d ; 0 )

•

b - l =1

Truoc het ta chung minh tinh chat sau day:
"Cho hinh vuong ABCD, cac diem M,N,P,Q Ian luot nam tren cac duang
thSng AB, BC, CD, DA. Khi do MP = NQ

MP 1 N Q ".

=^B(0;0),D(2;0)

flA = IB = l

ID = IA=1

b = 0,b = 2
d = 0,d = 2

hoac B ( 2 ; 0 ) , D ( 0 ; 0 ) .

V a y A ( l ; l ) , B ( 0 ; 0 ) , C ( i ; - l ) , D ( 2 ; 0 ) hoac

A(1;1),B(2;0),C(1;-1),D(0;0).

Chung minh: Ve ME 1 CD, E € CD; N F 1 A D , F e A D .
33


Phuong fihdp gidi Todn Hinh hoc thee chuyen de- Nguyen Phil Khdnh, Nguyen Tat Thti

Cty TNHH MTV DWH Khang Viet

Bai 1.3.2. Trong mat phang toa do Oxy cho dirong tron

Hal gia tri ciia b tuong I'mg toa do hai diem B va D
Vi AB CO he so goc duong nen B(5;0), D(l;2)

(C):x^+y^-8x + 6y + 21=0 va duong thSng ( d ) : x + y - l = 0 .

= > A B : x - 3 y - 5 = 0, A D : 3 x + y - 5 = 0.

Xac dinh tga do cac dinli cua hinh vuong ABCD ngoai tiep (C) biet A e (C)

Bai 1.3.5. Trong mat phang voi he toa do Oxy, cho hinh chii nhat ABCD c6

Jiuang ddn gidi

canh: A B : x - 3y + 5 = 0, duong cheo: B D : x - y - l = 0 va duong cheo AC qua

Ta CO I(4;-3),R = 2 Ian lugt la tarn va ban kinh cua (C).

/
01
oc

Ta CO toa do ciia B la nghiem cua he:

=4=>Xo =2,Xo=6

iH

-,2

=2V2o(xo-4)

D € BD

Goi B(X(j;X() - 7 ) e d ' .
2; X Q

=

6 ^

=:> A D n AB = A :

uO

=

B(2; -5), D(6; - 1 ) .

Vay toa do cac dinli cua hinh vuong la A ( 2 ; - l ) ; C ( 6 ; - 5 ) ;

B(2;-5),D(6;-1)

x= 4

AB = BC

(x-3)%y2=5

y = -2

*

Voi C i ( 4 ; - 2 ) = > D i ( 2 ; - 3 ) .

*

Voi

x=2

y =2

ww
w.

C2(2;2)=^D2(0;1).

lA = k.MI

^

7d-28

-d + 4

d + 22

d-2

Jiuong ddn gidi

2d + 7

*

<»d = - l ; d = 4.

.3 1,
d - - l = > D ( - l ; - 2 ) , A ( - 2 ; l ) va I ( | ; | ) ^ C ( 5 ; 0 )

Vay A ( - 2 ; l ) , B ( 4 ; 3 ) , C ( 5 ; 0 ) , D ( - l ; - 2 ) .

Jiu&ng ddn gidi

Gia su duong thSng A B qua M va c6 vec to phap tuyeh la fi(a; b)
7i 0) suy ra vec to phap tuyeh ciia B C l a : n j ( - b ; a ) .

=^ lA = (-1;-2) =^ lA = Vs =^ IB = ID = >/5

Do A B C D la hinh vuong nen d (P; A B ) = d (Q; B C )

b =2

'
^, ,

B C CO dang: - bx + ay + 4b + 2a = 0

Vi B £ B D n r > B ( 5 - 2 b ; b ) r r > i B - ( 2 - 2 b ; b - l )
1 c:>

j

lap phuong trinh cac canh ciia hinh vuong.

Phuong trinh A B c6 dang: ax + by - 2a - b = 0

(b -1)2

^

M ( 2 ; l ) , N ( 4 ; - 2 ) ; P(2;0); Q(1;2) Ian lupt thupc canh A B , B C , C D , A D . Hay

Vi A C l B D r ^ A C : 2 x - y - 5 = 0
Goi I la tarn ciia hinh vuong => I = A C n BD =:> 1(3; 1)

r:> IB^ = 5 o (2b - 2)2 + (b -1)2 = 5

'

Bai 1.3.6. Trong mat phang voi he toa do Oxy cho hinh vuong A B C D biet

(a +b

b =0

d +4 d+2

d = 4=>D(4;3) = B loai

Bai 1.3.4. Viet phuong trinh canh AB( AB c6 he so'goc duong), AD cua
hinh vuong ABCD biet A (2; - 1 ) va duong cheo BD: x + 2y - 5 = 0.

6d-4

Ta
up

ro

hoac

fa
ce

2 ( x - 3 ) + l.y = 0

bo
ok

Tii giac ABCD la hinh vuong suy ra :

3x + y - 4 d + l = 0'

A

Vi A, I , M thang hang nen ta c6:

/g
.c
om

Ggi C ( x ; y ) . K h i d 6 AB = (2;l);B(: = ( x - 3 ; y ) .
ABIBC

s/

Bai 1.3.3. Biet A ( 1 ; - 1 ) , B ( 3 ; 0 ) la hai dinh ciia hinh vuong ABCD . Tim toa
dp hai dinh C,D .

x-3y+5=0

Goi I la tarn cua hinh chu nhat =^ I la trung diem ciia BD => I

va cac hoan vi A cho C, B cho D .

Jiuang ddn glad

B(4;3).

y = 3^

D(d;d -1) => phuong trinh A D : 3x + y - 4d +1 = 0 .

ie

= 2V2 =i> XQ

x=4

iL

4)2

x-y-l=0

<=> {

B C l AB =:> BC: 3(x - 4) + (y - 3) = 0 cj. 3x + y -15 = 0 .

Duong thang d ' : x - y - 7 = 0 di qua tarn I va vuong goc voi d .

+ (X(, -

x-3y+5=0

Da

A(2;-l);C(6;-5).

=^ IB = ^/(Xo - 4 ) 2

JC

J-luang ddn gidi

. Goi A ( X ( , ; - X o +1) e d.

IA = ^ ( X ( , - 4 ) 2 + ( - X o + 4 ) 2

.i.iJ'.

hi

l A = V2R = 2^2

^

diem M ( - 9 ; 2). Tim toa do ciic dinh cua hinh chir nhat.

, hinh vuong ABCD ngoai tie'p duong tron nen

nT

Ta CO led

* -i':

Hay

-b

3b + 4a

b = -2a

^£27

Va2+b2

b = -a

'

35


va giao diem I ciia hai duong cheo nam tren duong thSng y = x . Tim toa do
dinhCvaD

b = -a . Khi do
A B : - x + y + l = 0 ;BC: - x - y + 2 = 0

,
Jiic&ng ddn gidi

A D : - x - y + 3 = 0 ; C D : - x + y + 2 = 0.

Ta c6: AB = (-1;2) => AB = N/S . Phuong trinh cua AB la: 2x + y - 2 = 0.

Bai 1.3.7. Trong mat phang voi h§ toa do Oxy cho ba diem I(1;1),E(-2;2),

Ie(d):y = x=^l(t;t).

F ( 2 ; - 2 ) . T i m tpa do cac dinh ciia hinh vuong ABCD, biet I la tam cua hinh
I la trung diein

Jiucmg dan gidi
Duong thang AB c6 phuong trinh dang: a(x + 2) + b(y - 2J = 0

Da

hi

nT

a - 3b

<=> a = - b

Ta

^ b ^

up

s/

Suy ra phuong trinh A B : x - y + 4 = 0, C D : x - y - 4 = 0.

ro

Phuong trinh BC va DA c6 dang x + y + c = 0

.c
om

/g

= 2 ^ = > c = 2,c = - 6 .

Vay toa do ciia C va D la: C 3 ' 3 ,

bo
ok

• BC:x + y - 6 = 0, D A : x + y + 2 = 0.Suy ra A{-3;1), B(l;5), C(5;l), D ( l ; - 3 ) .

ww
w.

Jiit&ng ddn gidi

fa
ce

diem M(2;l). Tim tQa do cac dinh cua hinh chu nhat

Ta CO BD n AB = B(7; 3), phuong trinh duong thSng BC: 2x + y - 17 = 0
Do A e AB=>A(2a + l;a), Ce B C C ( c ; 1 7 - 2 c ) , a ^ 3 , c ^ 7 ,
Suy ra I =

''2a + c + l a - 2 c + 17^
la trung diem cua AC, BD
1 2

2

,

M a l € B D o 3 c - a - 1 8 = 0 o a = 3c-18: A(6c-35;3c-18)
M , A, C thang hang <=> M A , M C ciing phuong

8 2

5_8)
^8.2^
t = -r^C
3'3
U'3.

3
t = 0=:>C(-!;0),D(0;-2)

,D

hoac C ( - l ; 0 ) , D ( 0 ; - 2 ) .

3'3

Goi M la trung diem canh BC, N la diem nam tren canh CD sao cho CN = 2ND.
Gia su M

^11

n va A N : 2x - y - 3 = 0. Tim toa do diem A .

2 ' 2
Jiunng ddn gidi

Gia sir hinh vuong A B C D c6 canh la a. Khi do, theo de bai, ta c6

Bai 1 . 3 . 8 . Trong mat phang voi h§ toa do Oxy, cho hinh chii nhat ABCD c6
canh AB: x -2y -1 =0, duong choo BD: x- 7y +14 = 0 va duong cheo AC d i qua

,D

• ' - i -'

Bai 1 . 3 . 1 0 . Trong mat phang voi he toa do Oxy, cho hinh vuong ABCD.

• BC : x + y + 2 = 0, D A : X + y - 6 = 0 . Suy ra A(l;5), B(-3;l), C(l;-3), D(5;l)

i

<=>

uO

iL

ie

a(x - 2) + b(y + 2) = 0 o ax + by - 2a + 2b = 0.

x+2

4

I6t-4I
Ngoai ra: d ( C ; AB) = C H o — ^ = -j=
V5
v5

Duong thang CD c6 phuong trinh dang:

d(I,BC) = d(I,AB) = 2V2:

AC v a BD nen ta c6: C(2t - l ; 2 t ) , D(2t;2t - 2 )

Mat khac: SJ^J^^D = AB.CH - 4 (CH: chieu cao) => C H =

«> ax + by + 2a - 2b = 0 voi a^ + b^ > 0.

3a-b

CLia

iH

vuong, AB di qua E va CD di qua F .

Vi d(I,AB) = d(I,CD):

^^^•.>P^A.^:':or.

/

•

Bai 1.3.9- Cho hinh binh hanh ABCD c6 di?n tich bang 4. Biet A ( l ; 0), B(0; 2)

b = -2a suy ra phirong trinh cac canh can tim la:
A B : x - 2 y = 0 ; CD : x - 2 y - 2 = 0; BC: 2x + y - 6 = 0; A D : 2x + y - 4 = 0.

01

•

Cty TNHH MTV DWH Khang Viet

Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, NguySn Tat Thii

oc

Phuongphdpgidi

A N = \/DA^TDN^ =
AM =

\/AB^ + B M ^

=

N M = VcN^TcM^ =

^2
a 2 +•a
9

TiOa
3

a 2 + aa^ ^ 4a^
' 4 ^ 9

5a
6'

Ap dung dinh ly cosin cho tam giac A N M , ta c6:
. A N ^ + A M ^ - N M ^ V2

cos N A M =
= —.
2-AN-AM
2
Do do, phuong trinh duong thang A M qua M va tao voi A N mot goc

Suyra
^

- 13c + 42 = 0 o ['^ =
[c = 6

Voi c = 6, ta c6: A ( l ; 0), C(6; 5), D(0; 2), B(7; 3).
36


Phuang phdpgiai Todn Hinh hgc theo chuyen de- Nguyen Phi'i Khdnh, Nguyen Tat Thu

Cty TNHH MTV DWIl

Gia sir duang thang A M c6 phap vector la ri = (a, b) (a^ +b'^ ^ 0). Khi do,

Vay CO hai diein can tim la: A

2a - b
ta ti'nh du-oc cos N A M =

g^j

VsVa^+b^'

Tu day, do cos N A M = — nen

bangiR-

/

01
oc

iH

Da

tuc

ie

iL
Ta

Bai 1.3.13. Trong mat phang Oxy cho hinh tlioi ABCD, phuang trinh hai

/ i

3' 3

[

1

Do A D = - nen A H = - hay:
3
3 ^
2

!-Vit
38

2

2

+

3

= i«3t2
9

ww
w.

fa
ce

Nell cho AB = X ta c6 BC = CD = 2x de dang ta thay

bo
ok

2

.c
om

Jiu&ng dan gidi

HAJ^f2i--;t-~
3
3

s/
up

canh AB, A D Ian lugt c6 phuang trinh x + 2y - 2 = 0 va 2x + y +1 = 0. Diem
M(l;2) nam tren canh BD. Tim toa do cac dinh cua hinh thoi.
Jiu&ng dan gidi

/g

Tim toa do cua diem A.

Goi H la hinh chieu cua M len A D ta c6 H

'

Vay phuang trinh AB la: x - y + l = 0 hoac x - 3 y +11 = 0 .

ro

A D : X - y V 2 = 0 . Trung diem M cua BC c6 toa do M ( l , 0). Biet BC = CD = 2AB..

Goi A ( V 2 t ; t ) , s u y ra

,;

hi

0,

2 J

Bai 1 . 3 . 1 1 . Cho hinh thang vuong ABCD, vuong tai A va D. Phuang trinh

A D = BE = V A B ^ - C E ^ = VSX = -

; ;

nT

0=

Vay CO tat ca hai diem A thoa man yeu cau de bai la : A(4, 5) va A ( l , - 1 ) .

Suy ra

0 voi a + b > 0 '

4{a-3b)(a-b)
a-fb^

Di?n tich ciia hinh chu nhat: S = d(P, AB).d(Q, BC) =
a= -1,b = l
(a - 3b)(a - b)
Ma S = 16 nen ta c6:
= 16=>
a ^ A b . l
a^+b^
3

X - 3y - 4 = 0. Vai ke't qua nay, ta tim duoc A ( l , - 1 ) .

X =•

2

uO

i r -3- ^

I

MH

2

Phuang trinh BC : b(x - 6) - a(y - 5) = 0.

(X
• V o i b = -3a : Chon a = 1, b = -3, ta c6 A M : 1 •

1

'

Phuong trinh AB CO dang: a(x - 4) + b ( y - 5 )

Tu day de dang tim dugc A(4, 5).

3x

i,,if'<')

Jiu&ng dan gidi

= 0, tiVc 3x + y - 1 7 = 0.

1-

1.3.12. Trong mat phang Oxy cho bon diem M(4;5), N(6;5), P(5;2),

thSng AB, BC, CD, DA Ian luxit di qua M , N , P, Q va dien tich hinh chir nhat

• Voi a = 3b : Chon b = 1, a = 3, ta c6 A M :
X --

3j2±S

Q(2;l) • Viet phuong trinh canh AB cua hinh chir nhat ABCD biet cac duong

ta c6

A/2 |2a - b| = Vs Va^Tb^ o 3a^ - 8ab - 3b^ = 0 <::^ a = 3b v b = -3a.
11

6±V6

Khnng Viet

Toa do ciia A la nghiem cua he:

"
_ _4

x+2y-2=0
2x+y+l=0

3
y =•

'3'3,

Gpi a la canh ciia hinh thoi, ta suy ra:
SABCD = 2SABD = 2 ( S A M B + S A M D ) =

A B ) + d(M, A D ) ] =

8a

~

Mat khac, S^^Q^ - AB.AD.sinBAD = a sin a
Ma cos a = cos( AB, AD) = - =:> sin a = - => S^J^Q^ =

5
, 3a^
Do do ta c6:
5

8a

S

=>a =

5

—

5

sVs

B ( 2 - 2 b ; b ) , D ( d ; - 2 d - 1 ) . T u AB = A D = - ^ ta tim dugc b = - l , d = -4
3
U 13^
Vay B(4;-l), D(-4;7) va C
3' 3
39

Cty TNHH MTV DWli

Phuong phapgiai Todn Hinh h(fc thco chuyen de- Nguyen Phu Khdnh, Nguyen Tai Thu

§ 4 . C A C B A I T O A N V E Dl/CfNG T R O N V A C O N I C

Gia su phuong trinh duong tron: x^ +
a-c = l

Khi giai cac bai toan ve duang tron chiing ta can luu y:

|'a = - l

• a - 2b + c = -5 o <^ b = 1 .

tron

a + b + c = -2

Cho hai duong tron (C,) c6 tarn I , , ban kinh Rj va duong tron (Cj) c6

. ;

'•Yi

[c = -2

Phuong trinh duong tron: x ^ + y ^ - x + y - 2 = 0.

/

, ban kinh Rj . Khi do, ta c6 cac ket qua sau:

01

tarn

+ ax + by + c = 0 .

Ba diem M , N , H thuoc duang tron nen ta c6 he phuong trinh :

1. J^fhdm cdc bdi todn lien quan den du&ng tron.
1) Vi tri turnip doi ^iim hai dican^

Khang Vic,

Vidu /.4.2.Trong mat phang voi h^ toa do Oxy, cho cho hai diem A(2;0)

oc

• (C,) va (C2) khong CO diem chung khi va chi khi

va B(6; 4 ) . Viet phuong trinh duong tron ( C ) tiep xuc voi true hoanh tai A

iH

I j l 2 > R , + R 2 hoac I , l 2 < | R , - R 2 •

Da

va khoang each t u tam cua ( C ) den diem B bang 5.

(Cj) va (C2) tiep xuc ngoai khi va chi khi I j l 2 = R i + R 2 -

•

(Cj) va (C2) tiep xiic trong khi va chi khi I,l2 = Rj - R 2 •

Goi I(a;b) va R Ian luot la tam ciia va ban kinh cua ( C ) .

(c ; ^ r ; t

•

(C,) va (C2) c^tnhau khi vachi khi | R , - R 2 | < I , l 2 < R i + R 2 -

Vi ( C ) tiep xiic voi Ox tai A nen a = 2 va R = b

J ( - <*

2)

Vi tri turnip doi ^im ditxtn^ than;^ va dinrnif

Matkhac: IB = 5 o 4 2 + ( b - 4 f = 5 ^ ^ b = l , b = 7

ie

uO

nT

hi

•

iL

tron

Ta

Cho duong tron (C) c6 tam I , ban kinh R va duong thang A . Goi H la hinh

s/

chieu cua I len A va dat d = I H = d(I, A). Khi do:

up

• (C) va A khong c6 diem chung khi va chi khi d > R .

ro

• (C) va A CO diing mot diem chung khi va chi khi d.= R . Luc nay A goi la

/g

tiep tuyen cua (C), H la tiep diem.

.c
om

Chu y: Tir mot diem M nSm ngoai duong tron (C) luon ve duoc hai tiep

bo
ok

tuyeh M A , MB (A,B la cac tiep diem) den (C). Khi do M A = MB va I M la
phan giac ciia goc A M B .

fa
ce

• (C) va A CO diem A,B chung khi va chi khi d < R . Khi do H la trung diem

ww
w.

cua AB va ta c6 cong thuc R^ = d^ + -^^^.

Vidu 1.4.1. Trong mat phang voi he true toa do Oxy, cho tam giac ABC c6

•

•

2

Voi b - 1 thi phuong trinh duong tron ( C ) : (x - 2) + (y - 1 )

Voi b = 7 thi phuong trinh duong tron ( C ) : (x - i f

2

+ (y - i f

H,M,N.
Xgigidi.

(BHIAC
<

40

HeAC

f4(x + 2 ) - 4 ( y + 2) = 0
<=> <

l4x + 4 ( y - 2 ) = 0

fx = l
<=> <

y=l

H(l;l)

'

'

= 49.

A j : 4x - 3y - 24 = 0, A j : 4x + 3y + 8 = 0. Viet phuong trinh duong tron ( C ) di
qua M va tiep xuc voi hai duong thang A j , A j .
JCffigidi.
Gpi I(a; b) la tam va R la ban kinh ciia duong tron (C).

'Jih i ;

.

Vi ( C ) tiep xiic voi hai duong th^ng Aj va A2 nen ta c6 d(I,Aj) = d(I,A2)
Hay

4a-3b-24

4a + 3b+S

= Ro

4 a - 3 b - 2 4 - 4 a + 3b + 8
4a-3b-24 = -4a-3b-8

a = 2, phuong trinh ( C ) : (x - i f

2

« b = 3, b -

25
Suy ra phuong trinh ( C ) : (x - 2)^ + (y - 3)^ = 25
hoac

<=>

a =2

+ (y - b ) ' . (^b +16)
25

Do M e ( C ) nen (6 - 2 ^ + (6 - bf = i^^ll^

Taco M ( - l ; 0 ) , N ( l ; - 2 ) , A C = ( 4 ; - 4 ) . G 9 i H ( x , y ) , t a c6:

= 1.

r-

Vi du 1.4.3. Trong mat p h i n g Oxy cho diem M(6;6) va hai duong thang

A(0;2),B{-2;-2), C(4;-2).Goi H la chan duong cao ke t u B; M , N Ian lugt
la trung diem cua AB, A C . Viet phuong trinh duong tron di qua cac diem

>

(C):(x-2)^.(y-f)2=^
4
16

87
4

4-


Cty TNIUl M'lV DWH Khang Viet

Phumtg phdpgiai Todn Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Td't Tim

16
• b = - — , p h u o n g t r i n h cua (C): (x - a)^ + V + —
^
3
3
D o M e (C) nen (6 - a)^ + 6 + —
3 j
4. T r o n g mat phang

( C ) : x^ +

p h u o n g t r i n h v 6 nghiem.

Suyra AB^ = 5(Xi - X 2 ) ^ = 5[(x, + X j ) ^ -4x,X2^
X) + X 2 _

v o l he toa do O x y , cho d u o n g t r o n

- 2x - 2y + 1 = 0 va d u o n g thang d : x - y + 3 = 0 . Viet p h u o n g

y^ = X i + X 2 + — = •

12 27^
5

10

2 " 10

Da

nT

uO

d ^ ( I , A B ) . ^ =I M ^ o ( l ^ . ^

;

<=> (a - 1 ) ^ + (a + 2)^ = 9 o a^ + a - 2 = 0 o a = l , a = - 2 .

,, ij "rm I

iL
Ta
s/

J^I'M i s T

Vay p h u o n g t r i n h cua (C): (x - 5)^ + ( y +1)^ = 74.

C/iuiy.-Ngoai each giai tren, ta c6 the sir d u n g c h u m d u o n g tron de giai. Cu the:
Vi (C) d i qua cac giao d i e m cua ( C j ) va (C2) nen p h u o n g t r i n h cua (C)

up

+{y-lf=4.

ro

1.4.5. T r o n g mat phang O x y cho d u o n g t r o n ( C ^ ) : x ^ + y ^ - 2 x - 2 y - 1 8 = 0

CO dang: m ( x ^ + y^ - 2x - 2y - 1 8 ) + n(x^ + y^ + 2x - 4y - 3) = 0 .

/g

+ (y - 2)^ = 8. C h u n g m i n h rang hai d u o n g tron

Do (C) d i qua M ( 0 ; 6 ) n e n t a c6: 2 m + 3n = 0 , ta chon m = 3,n = - 2

t r o n (C) d i qua ba d i e m A , B, M ( 0 ; 6).

K h i do p h u o n g t r i n h (C): x^ + y^ - lOx + 2y - 48 = 0 .

ijpT gidi.

fa
ce

D u o n g t r o n ( C j ) c6 t a m I j ( l ; l ) , ban k i n h R j = 2 V 5 .

bo
ok

.c
om

(Cj) va (C2) cat nhau tai hai d i e m phan biet A , B. Viet p h u o n g t r i n h d u o n g

==275 + 2V2

ww
w.

Do 2N/5-272=RJ-RI
Vi du 1.4.6. T r o n g he toa do O x y , cho d u o n g t r o n ( C ) : (x - 6)^ + (y - 2)^ = 4 .
Viet p h u o n g t r i n h d u o n g t r o n ( C ) tiep xuc v o i hai true tga do O x , O y dong

D u o n g t r o n ( C j ) c6 t a m l 2 ( - l ; 2 ) , ban k i n h R2 = 2N/2 .

cat nhau tai hai d i e m phan biet A , B .

,

= (2a.3)^.(a.6)^c.a.l

Suy ra 1(5; - 1 ) , ban k i n h R = I M = 5^2 .

• a = l=>r(l;4)=>(C'):(x-l)^+(y-4)2 =4

•

s ,

oc

;

>;,>].,{

ie

V i (C) va ( C ) tiep xiic ngoai v o i nhau nen 11' = R + R' = 3

Matkhac:

hi

Goi I la tam cua d u o n g t r o n (C), suy ra I e A => I(2a + 3; - a )

r la tam va R' la ban k i n h ciia d u o n g t r o n ( C ) ta c6 R' = 2R = 2 v a
r(a;a + 3)

va d u o n g t r o n ( € 3 ) : (x + if

^

P h u o n g trinh d u o n g t r u n g true A cua doan AB: x + 2y - 3 = 0 .

D u o n g t r o n (C) c6 tam 1(1; 1), ban k i n h R = 1 .

Vidu

M

iH

JCgigidi.

• a = - 2 = > r ( - 2 ; l ) = ^ ( C ' ) : ( x + 2)^

2;

P h u o n g t r i n h d u o n g thSng AB: 4x - 2y +15 = 0 nen

tron (C) va tiep xuc ngoai v o i d u o n g t r o n (C).

Goi
ed

X2;2x2 +

12
5
15_27

Goi M la t r u n g d i e m A B , suy ra

t r i n h d u o n g t r o n ( C ) c6 t a m M tren d, ban k i n h bang 2 Ian ban k i n h d u o n g

r

,B

Goi x,,X2 la hai n g h i e m cua (*), suy ra A | ^ x , ; 2 x i + y

/

dul.4.

25

25

01

Vi

(4a-8)^

15

f

(4a-8)^

J h o i tiep xiic ngoai v o i (C).
nen ( C J ) va (C2)

Xffigidi.
D u o n g tron (C) c6 tam I (6; 2 ) , ban k i n h R

Toa d p giao d i e m ciia ( C j ) va (C2) la n g h i e m ciia he:
x^ + y ^ - 2 x - 2 y - 1 8 = : 0

x^+y2-2x-2y-18 =0

{x + lf

x^ + y^ + 2 x - 4 y - 3 = 0

2.

Goi ( C ' ) : ( x - a f + ( y - b f = R ' 2 t h i ( C ) c6 tam I ' ( a ; b ) , ban k i n h R ' .
V i ( C ) tiep xiic v o i Ox, O y

+{y-2f

=8

n e n s u y ra d ( r , O x ) = d(r,Oy)<=>|a| = |b| = R'<=>

42

x^ + y 2 - 2 x - 2 y - 1 8 = 0

y = 2x.l^
^

2

2x.l^ =y
2 ^

5 x 2 + 2 4 x + — = 0 (*)
4

H o n n u a ( C ) tiep xiic v o i O x , O y va tiep xiic ngoai v o i (C) nen ( C ) n a m

93

f

en p h a i true O y , d o do a > 0 .
43


Phuang phiip gidi Todn Hinh hoc theo chuyen dc- Nguyen Pltii Khduh, Nguyen Tat Thu

Cty TNIUI MTV DVVII Kliang Viet

T H l : a = b = R = > ( C ' ) : ( x - a f + ( y - a f =3^

API = 30" ^ IP = 2IA = 2R = 6 .

Vi (C) tiep xuc ngoai voi (C) nen:

Suy ra P thupc vao duang tron (C)
CO tam I va ban kinh R' = 6 .

Ma P e d nen P chinh la giao diem
ciia duong thing d va duong tron (C)
Suy ra tren d c6 duy nhat diem P thoa
man yeu cau bai toan khi va chi khi duong
thang d tiep xuc voi duong tron (C) tai P

^a-2
a = 18

/

Trirong hop nay c6 2 duong tron la :

a = -b = R => ( C ) : (x -a)^ + (y + a)^ = a^

hay la d(I,d) = 6

oc

TH2:

01

( C ; ) : ( x - 2 f + ( y - 2 f =4 va ( q ) : {x - 1 8 ) ^ (y - i s f = 18^

m = 19,m = - 4 1 .

iH

i r = R + R ' o i J ( a - 6 ) ^ + ( a - 2 ) ^ = 2 + a<r>

Vi du 1.4.9. Cho duong thang A : x + y + 2 = 0 v a duong tron

i r = R + R ' » ^(a - 6)^ + (a + 2 ^ = 2 + a o a = 6

(C): x^ + y^ -4x -2y - 0. Gpi I la tam va M thuoc duong thang A . Qua M ke

nT

hi

Da

Tuong tu nhu truong hop 1, ta CO :

tiep tuyen MA,MB. Tim M sao cho di§n tich tu giac MAIB hang 10.

uO

Vay truong hgp nay c6 1 duong tron la (C3 j : (x - 6)^ + (y + 6)^ = 36 .

ie

(De thi DH Khoi A JCffigidi.

iL

Tom lai, c6 3 duong tron thoa can tim la :

up

Vi du 1.4.7. Trong mat phSng Oxy cho duong tron (C): (x -1)^ + (y - 2)^ = 9

s/

Ta

( x - 2 f + ( y - 2 f =4, (x-18)^+(y-18^=182 va ( x - 6 ) ^ + ( y + 6 f =36.

2011).

Duong tron (C) c6 tam 1(2; 1), ban kinh R = yl5=i'Al = S .
Matkhac

S^MAI = 2SAIBM

=5

.c
om

/g

tiep tuyen MA, MB (A,B la tiep diem). Tinh dien tich ciia tu giac MAIB .

ro

CO tam I va diem M(5;-3). Chung minh rang tu M, ta c6 the ve den (C) hai

JCgigidi.

bo
ok

Duong tron (C) c6 tam 1(1; 2), ban kinh R = 3.

SMAIB = 2 3 . ^ , 1

= lA.MA = R . N / M I ^ - R ^

= 3.V41-9

=12^2 (dvdt).

ww
w.

Ta CO

fa
ce

Vi MI = N/41 > R nen M n3m ngoai duong tron (C), do do tu M ta luon ve
duoc hai tiep tuyen toi duong tron (C).

Vi du 1.4.8. Trong mat phang voi he tga do Oxy, cho duong tron

= * i M A . I A = 5 r ^ M A = 2^/5
2

Suy ra IM^ = lA^ + AM^ = 2 5 .
Ma M e A nen

suy ra IM^ = 25

M(m;-m-2),

(m - 2) + (m + 3)^ = 25

o m ^ + m - 6 = 0<=>m = -3,m = 2.
V^y M(2;-4) va M(-3;l) la hai diem can tim.
Vi du 1.4.10. Trong mat phSng Oxy, cho duong tron (C): (x - 4)^ + y^ = 4 va
diem E(4;1) . Tim tpa dp diem M tren tryc tung sao cho tu M ke dupe hai

(C): (x -1)^ + (y + 2)^ = 9 va duong thing d : 3x - 4y + m = 0 . Tim m de tren

tiep tuyen MA, MB den duong tron (C) voi A,B la hai tiep diem sao cho

d CO duy nhat mpt diem P ma tu do c6 the ke dupe hai tiep tuyen PA,PB

duong thSng AB di qua diem E.

toi (C) (A,B la cac tiep diem) sao cho tam giac PAB deu.
JCffigidL

Duong tron (C) c6 tam va ban kinh Ian lupt la: 1(1;-2); R = 3 .
Do tam giac PAB deu nen
44

JCgigidi.

Duong tron (C) c6 tam 1(4;0), ban kinh R = 2.
Gpi M(0; m), gia su T(x; y) la tiep
45


Cty TNHH MTV DWH

Toiiit Hhth hoc theo chuyat de- Ngmjcn Phi't Khiinh, Nguyen Tat Thu

trinh tiep tuyeh chung ciia (C^) va ( C 2 ) tai A. Go! d la mot tiep tuyeh

Suy ra M f = ( x ; y - m ) , I T = ( x - 4 ; y ) .

chung ciia (C,) va ( C 2 ) khong di qua A, duang thSng d cat duong thang

x^ +y^ - 8 x + 12 = 0

rio'i hai tarn tai B. Tim toa dp diem B.

x^ +y^ - 4 x - m y = 0

=>4x-my-12 = 0

Duong tron (Cj) c6 tam 1(3; 2) va ban kinh R = 3.

M

Do do, phuong trinh duong thang

AB:4x-my-12 = 0

/

01

AI'

AI

Vi du 1.4.11. Trong mat phang voi he toa do Oxy,cho duong tron

uO
ie
iL
s/

Duong tron ( C ) c6 tarn 1(1;-2), ban kinh R = N/S .

up
ro

/g
.c
om

^|x2+y2-2x„+4yo=0

• (m - l)xy + (m + 2)yQ + m = 0 .

>
fa
ce

Suy ra phuong tiinh A B : (m - 1)x + (m + 2)y + m = 0.

3

>/2V(m-l)-^+fn.+2)2

vlO

ww
w.

Mat kliac AB iiv\i d mpt goc cp voi coscp = — ^ nen ta c6:
Vio
|m-l-m-2;

=

27
=>A =

27
5'5j

Tiep tuyeh chung ciia (Cj) va (C2) tai A.

—2——~T

0 m^ + m ^ 0 <x> m = 0,m = - 1
Thu lai ta thay ca hai truong hop nay ta deu I M = R hay M e (C).
Vay khong c6 diem M thoa yeu cau bai toan.
Vi dif. 1.4.12. Cho hai duang tron
( C i ) : (x - 3)' f (y - 2 ) ' = 9 va (C^): (x - 7 ) ' + (y + 1 ) ' = 4 .
hung minh (C,) va (Cj) tiep xiic ngoai voi nhau tai A. Vie't phuong

>;;

Vec to phap tuyeh cua tiep tuyeh tai A : n = 11' = (4; -3)

; ,^

,

Phuong trinh tiep tuyeh chung ciia (Cj) va (C2) tai A la: 4x - 3y - 21 = 0.
,

Gpi B(xQ;yQ), theo gia thiet ta c6

bo
ok

<=> s

X

Ta

Jdffi gidi.

+ (>'{) + 2)(yo - m) = 0

lA

R
R
x-7 = -±(x-3)
y +l = -|{y-2)

3

Gpi M ( m ; m) va T ( X Q ; y(,) la tiep diem ve tir M de'n (C). Khi do, ta c6

R'r^

nT

M tren duang thang d, biet t u M ke duoc hai tiep tuyeh M A , M B den (C) (A,
B la cac tiep diem) va duang thang AB tao voi d mot goc 9 vai cos (p =

R'

. =_c:>rA =

hi

(C): x'^ + y'^ - 2x + 4y = 0 va duang thang d : x - y = 0 . Tim toa do cac diem

T6(C)

,

oc

B

Vay M(0;4) la diem can tim.

I(>^o - l)(>^o -^)

,, ,

Goi A ( x ; y ) . Theo gia thiet ta c6:

Nen AB di qua E<=>16-m-12 = 0 o m = 4.^

IT.MT = 0

^

Duong tron ( C 2 ) C O tam r(7;-1) va ban kinh R' = 2.

iH

^
, fTe(C)
Ta co:

MT.IT = 0

Khang Viet

Da

Phumtgphdpgitii

B T ^ R ;
B I

Suy ra IB' = — I B o
^

R

R

Xo-7 = -(xo-3)
yo+i-f(yo-2)

Xo=15
yo=-7'

•B(15;-7).

Vi du 1.4.13. Trong mat phSng vai h^ tpa dp Oxy, cho duong tron
( C ) : ( x - l ) 2 + ( y - l ) 2 =10. Duong tron ( C ) tam r ( - 2 ; - 5 ) cat (C) tai hai
_die'm A, B sao cho AB = l^Js . Viet phuong trinh duang thang A B .
Xffi gidt

Duong tron (C) c6 tam 1(1; 1), ban kinh R = VlO . Dp d a i l l ' = 3V5
Gpi H la giao diem cua IF vaAB
suy ra H la trung diem AB nen A H = Vi

''
47


Phuaiig phdp gidi Todn llxnh hoc theo chuyen de- Nguyen Phii Khdnh, Nguyen Ta't Thu

Cty TNHH MTV DWH Khang Vift

y i A di qua M nen phuong trinh A c6 dang: ax + by + 6a - 3b = 0

Do i r 1 AB nen ta c6: I H = VlA^ - A H ^ = y/s

Ma ta c6: H I = 1 => d(I, A) = 1 »-l^i^M = ^

T H 1: H thuocdoan 11" => I ' H = 2^5 =^ I H = - I T
3
Ma i H = ( x n - l ; y „ - l ) ,

Va + b^

n ' = (-3;-6)

» 1 5 b ^ -56ab + 48a^ =0<=>b = - a , b = —a
3 duong5thang can lap.
Vay A : 3x + 4y + 6 = 0 hoac A : 5x + 12y - 6 = 0 la

non ta c6:

n = - l i r = (l;2)

H

va

/

01

di qua

Vi du 1.4.15. Trong mat phang voi he toa do Oxy cho duong tron (C) c6
phuong trinh: x^ + y^ + 4x + 4y + 6 = 0 va duong thSng A : x + my - 2m + 3 = 0

nhan

oc

AB

H(0;-1)

=-1

voi m la tham so thuc. Gpi I la tam cua (C). Tim m de A cat (C) tai hai

iH

Vi

Vu

lam V T I ^ nen

diem phan biet A , B sao cho tam giac l A B c6 dien tich Ion nha't.

Da

=-2

nT

Duong tron (C) c6 tam I ( - 2 ; - 2 ) , ban kinh R =
Gpi H la hinh chieu cua I tren A .

ie

SUV ra r H - 4 N / 5 = i > i H = -!-ir

4

iL

De A cat duong tron (C) tai 2 diem A,B phan biet thi: I H < R

ro

up

yii - i = - 2

=>H(i;-i).
^4 2'

s/

1

Ta

1

= --

.c
om

/g

Phuong trinh A B : x + 2y + — = 0 .

Vi du 1.4.14. Trong mat phang Oxy cho duong tron (C): (x -1)^ + (y +1)^ = 9

Khi do S „ ^ 3 = ^ I H . A B = IH.HA < I M L L M A ! = ^

fa
ce

J[ffigidi.

=1

m =0
o l 5 m ^ - 8 m = 0<=>

_8_

m =•
15

tron (C) tai hai diem phan biet A, B sao cho tam giac lAB c6 dien tich bang
2N/2 va A B > 2 .

=^

Suy ra maxS^^i^g = 1 khi I H = H A = 1 < R
l-4m
= 1 <=> 1 - 4m = Vm^ +1 o 1 - 8m + 16m'^ = m'^ + 1
Vm^+1

bo
ok

C O tam I . Viet phuong trinh duong thang A d i qua M(-6;3) va cat duong

8
Vay C O 2 gia trj ciia m thoa man yeu cau la: m = 0 va m = — .

10

Vi

Gpi H la trung diem cua A B . Suy ra I H 1 A B

A : x + y - 2 = 0 va duong tron ( T ) : x ^ + y ^ - 2 x + 2 y - 7 = 0 . Chung minh rang

ww
w.

Duong tron (C) c6 tam 1(1;-1), ban kinh R = 3.

=>

SAAIB

= - H I . A B = 2>^ =^ A B -

2

—
HI

Matkhac:AH2+HI^=IA2
AB^

+ Hl2 = 9 o ^ - + Hl2=9
HI^

<=>Hl''-9Hl2+8 = 0<:>
48

4l.

uO

T H 2: H khong nam trong doan 11',

XH-1

J!gi gidi.

hi

phuong trinh AB la: x + 2y + 2 = 0.

Suy ra

10

A

H I = 1 =:> A B = 4V2

H I = 2N/2=>AB = 2 (loai)'

diJL

1.4.16. Trong mat p h i n g voi h§ tpa dp Oxy cho duong thang

A ck (T) tai hai diem phan bi?t A, B va tim tpa dp diem C tren (T) sao cho
tam giac ABC c6 dien tich bang (3 + 72)V7 .
Duong tron (T) C O tam 1(1;-1), ban kinh R = 3
Ta C O d(I, A) =

< R =i> A va (T) cat nhau t ^ i hai diem phan bi?t A, B

Va AB = 2^/R^-d^(I,A)=2^/7.

49