Phương pháp giải toán hình học theo chuyên đề (p1)

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TNHH
MTV
DVVH
Khang Viet
PHUONG
PHAP TOA DO
TRONG
IVIAT
PHANG
A, LY THUYET GIAO KHOA
I.
Tpa dp
trong
mat phang.
• Cho u(xpyj);
v(x2;y2)
va keR. Khi do:
1) u+v = (xi
+X2;yi
+y2)
2) u-v = (xi-X2;yi-y2)
3) ku=(kxi;kyi) 4) Z=Jx\+y\)
u=vc^r^
"''""''^
6)
U.V
=
X]X2
+y]y2=>ulv<;:>u.v
= 0<=>

\-^\2
+ y]y2 = 0
• Haivecta u(xj,yj);
v(x2;y2)
ciing phirang vai nhau <=>
• Goc giija hai vec to u(xj,yj); v(x2;y2):
U.V
XiX2+yiy2
cos(u,v)=
u
V
Cho A(x^;y^); B(xB;yB). Khi do :
1) AB = (xB-XA;yB-yA) 2) ^^=^3 = ^{x^ - x + {y ^ - y
3)
_XA+XB
I
~
trong
do I la
trung
diem ciia AB.
• AB 1 CD o AB.CD - 0
• Cho tarn
giac
ABC voi A{x^;y^), B(xB;yB), C{x^;y^). Khi do trong tarn
G(x(,;yg)
ciia tarn
giac
ABC la :
V

_XA+XB+XC
XG-
^
yG=
I
II.
PhirotTg trinh duong
thang
,, ,^,
1.
'Phuang trinh duong thdng
1.1. Vec to chi
phucmg
(VTCP), vec to
phdp
tuyen
(VTPT) cua
duong
thang:
Cho duong thang d.
• n = (a;b)
?t
0 goi la vec to phap tuyen cua d neu gia ciia no vuong voi d.
3
Phiam^
phiip giui loiin llinli hoc
Iheo
chuycn
de-
Nguyen

Phti
Khdnh,
Nguyen
Tat
Thti
•
u
=(uj;u2)^0
goi la vec ta chi phuong cua d ne'u gia cua no trung
hoac
song song voi duong thang d.
Mot
duong thang c6 v6 so
VTPT
va v6 so VTCP ( Cac vec to nay
luon
cung
phuong voi nhau)
•
Moi
quan he giua
VTPT
va
VTCP:
n.u = 0 .
•
Ne'u n = (a; b) la mpt
VTPT
cua duong thang d thi u =
(b;

-a) la mot VTCP
cua duong thang d .
•
~
!
(
5
f
•
Duong thang AB c6 AB la
VTCP.
1.2. Phuwig trinh dumig thang
1.2.1.
Phuatig trinh tong qudt cua duong thang:
Cho duong
thMng
d di qua diem
A(xQ;yQ)
va c6 n = (a;b) la
VTPT,
khi do
phuong
trinh
tong quat
ciia
d c6 dang: a(x -
XQ)
+ b(y - yp) = 0.
1.2.2.
Phuovg trinh

tham
so cua duong thang:
Cho duong thSng d di qua diem
A(xo;yo)
va c6 u = (a;b) la
VTCP,
khi do
X
= XQ
+
at
phuong
trinh
tham so cua duong thang d la: , t
G
R .
[y
=
y(,+bt
2. Vi tri tuang doi giua hai duang thdng.
Cho hai duong
thcing
dj
:
a^x + bjy + c^ = 0; d2
:
a2X + b2y +
C2
= 0
.

Khi do vi tri
|a,x
+
b,y
+
Cj
=0
tuong
doi giua chung phu thuoc vao so nghiem cua h^ : < , (I)
[a2X + b2y+
C2
=0
•
Neu (I) v6 nghiem thi d^ /
/d2
.
•
Ne'u (I) v6 so nghiem thi d^ = dj
•
Ne'u (I)
CO
nghiem duy nha't thi dj va d2 cat nhau va nghiem
ciia
he la toa
do giao diem. '
3.
Goc
giua hai
dijcang
thdng.

Cho hai duong thang dj
:ajX
+
b^y+
Cj
=0; d2
:a2X
+
b2y
+
C2
=0. Goi a
la
goc nhon tao boi hai duong thang dj va d2 .
Ta
CO
:
cosa
=
aja2
+
bjb2
^/a^Tb^
^/af+b
4.
JChodng
each tit mot diem den ducrng thdng.
Cho
duong th5ng
A

:
ax + by + c = 0 va diem
M(XQ;y^).
Khi
do khoang each
tu M den A dugc tinh boi
cong
thuc:
Cty
TNHH
MTV DWH Khang Viet
d(M,(A)):
axp
+
byp
+
c
Va^+b^
5.
(phuong trinh duang phdn gidc cua goc tao boi hai duang thdng
Cho hai duong thang d^
:
a^x + b^y + c^ = 0 va d2
:
ajX
+
b2y
+
Cj
= 0

Phuong
trinh
phan giac
ciia
goc tao boi hai duong thang la: -
,
v
, •
ajX
+
b^y
+
Cj
a2X
+ b2y +
C2
+
^/a^^b[ • ,i
c-i ;
; .
III.
Phuang trinh duong tron. rmu.j.
1.
<Phuang
trinh duang tron:
Cho duong
tron
(C) tam
I(a;
b), ban

kinh
R, khi do phuang
trinh
ciia
(C)
la:
(x-a)2+(y-b)2=R2.
Ngoai
ra phuong
trinh:
x^ + y^-2ax-2by + c = 0 voi a^+b^-oO cQng
la
phuong
trinh
ciia
duong
tron
c6 tam
I(a;b),
ban
kinh
R = Va^ + b^ -c .
2. Phuang trinh tiep tuyen:
Cho duong
tron
(C): (x-a)^+(y-b)^ =R^
•
Tiep tuyen A
ciia
(C) tai diem M la duong thang di qua M va vuong goc

vai
IM
.
•
Duong thang
A
:
Ax + By
+
C = 0 la tiep tuyen
ciia
(C)
<=>
d(I,
A)
= R '
•
Duong
tron
(C): (x -a)^
+
(y -
b)^
= R^ c6 hai tiep tuyen cung phuong voi
Oy
la x = a ± R . Ngoai hai tiep tuyen nay cac tiep tuyen con lai deu c6 dang:
y
= kx
+
m .

IV.
E
lip
1.
'i)inh
nghra Trong mat phang cho hai diem co'djnh
Fi,F2
c6
Y^Yj
=2c. Tap
hop cac diem M cua mat phang sao cho
MF^
+MF2
=2a (2a khong doi va
a > c> 0) la mot duong
elip.
•
F,,F2
:
la hai
tieu
diem va 2c la
tieu
cu
ciia
elip.
•
MF|,MF2
: la cac ban
kinh

qua
tieu.
2. Phuang trinh chinh tdc cua
elip:
4
+
4 =
^
voi b^=a^-c^. K'.
a
2
b^
Vay
diem
M(xo;y(,)
e (E)
•
=
1
va
<a
Yo
<b,
Phumtg phcip giiii Toan Hhih hoc
theo
chuyen tie- Nguyen Phu Khdnh, Nguyen Tat Thu
3. Tinh chat
v>d
hlnh dang cua
elip:

Cho
(E):
—
+
^ = 1, a>b.
a b
•
True doi xung Ox,Oy . Tarn do'i
xiing
O. j,
•
Dinh:
A[(-a;0),
A2(a;0), 6^(0;-b) va 62(0; b). A^A2 = 2a goi la do dai
true
Ion,
B]B2
= 2b goi la do dai true be.
•
Tieu
diem: F|(-c;0),
F2(e;0).
^
•
Noi tiep trong
hinh
ehir nhat co so
PQRS
CO
ki'ch

thuoc 2a va 2b voi b^ = a^ - e^.
Q
0
s
Bi
R
-ex(,.
•
Tam sai: e =
—
=
<
1
a a
a a^
•
Hai duong chuan:
X
=
±—=
±
—
e e
V.
Hypebol
1.
^inh nghia: Trong mat phang voi h$ toa do Oxy eho hai diem Fi, F2 eo
FjF2
=2c. Tap hop cac diem M
ciia

mat phSng sao eho
MF^
-MFj =2a (2a
khong
doi va c>a >0) la mpt Hypebol.
•
Fp
F2
:
la 2
tieu
diem va F|F2 = 2e la
tieu
eu.
•
1VIF[,MF2
:
la eac ban
kinh
qua
tieu.
2.
'Phimng
trinh chinh idc cua
hypebok
x^
y^
a^
=
1

voi h^=c^-a^.
3. Tinh chat vd hlnh dang cua
hypebol
(fi):
•
True doi xung Ox (true thuc), Oy (true ao). Tam doi xung O .
•
Dinh:
Aj(-a;0),
A2
(a;0).
DQ
dai true thuc: 2a va do dai true ao: 2b.
•
Tieu
diem Fi(-e; 0), Fj ( c;
O)
.
•
Hai
tiem
can: y =
±
—x
a
•
Hinh
eho nhat co so
PQRS
c6

kieh
thuoe 2a, 2b voi b^ = c^ - a^.
•
Tam sai: e
=
—
=
a
•
Hai duong chuan: x =
±—
=
±
—
Cty
TNHH
MTV DWH Khang Viet
•
DO
dai cac ban
kinh
qua
tieu
cua
M(xo;y(,)e(H):
+)
MF^
= ex„ + a va MF2
=eX(,
-

a
khi XQ > 0.
+)
MFj = -exp - a va MF2 = -exp + a khi XQ < 0 .
2
2 .2 2
X(,]>a.
.
M(xo;yo)6(H):\-J^
=
l«^-fj
=
l vataluonco
a D a b
VI.
Parabol
j.<^inhnghia:
Parabol la tap hop cae diem M cua mat phang
each
deu mot duong thang
A
co'dinhvamot diem F co
dinh
khong thuoe A.
A
: duong chuan; F :
tieu
diem va
d(F,A)
= p > 0 la tham so'tieu.

2.
'Phuxmg
trinh chinh tdc cua
^arabd:
= 2px
3.jrinh
dang cua Parabol
(<P):
•
True Ox la true do'i xung,
dinh
O.
Tieu
diem
F(^;0).
•
Duong chuan
A
:
x =
•
M(x;y)e(P):
MF = x + ^ voi x>0.
B,
CAC
BAI
THlfONG
GAP
§
1.

cAc
BAI
TO
AN
CO BAN
1.
Xg.p
phuang trinh duang
thang.
De lap phuong
trinh
duong thang A ta thuong dung cac
each
sau
•
Tim
diemM(xo;yo)
ma A di qua va mot
VTPT
n = (a;b). Khi do phuong
trinh
duong thang can lap la: a(x -
XQ)
+
b(y- yp) = 0 .
•
Gia su duong thang can lap
A
:
ax + by

+
e = 0
.
Dua vao dieu
kien
bai toan ta
tim
dugc a = mb,c = nb. Khi do phuong
trinh
A:mx + y + n = 0. Phuong
phap
nay ta thuong ap dung doi voi bai toan
lien
quan den khoang
each
va goe
•
Phuong
phap
quy
tich:
M(xQ;yQ)e
A:ax + by + e=^Oc:> axy + by^
+
e = 0 .
Vidu 1.1.1.Trong mat phSng voi he toa do Oxy cho duong
tron
(C):(x-])2+(y-2)2=25.
1)
Viet

phuong
trinh
tiep tuyen
ciia
(C) tai diem
M(4;6),
'
2)
Viet
phuong
trinh
tiep tuyen cua (C) xua't phat tu diem
N(-6;l)
Phucntg
phap
giai
ToAn
Ilinh
hoc
theo
chuycn
lic-
Nguyen
Pliii
Khanh,
Nguyen
Tat
Thii
3) Tu
E(-6;3)

ve hai tie'p tuye'n EA, EB (A, B la tie'p diem) den (C). Viet
phuong
trinh
duong thang AB.
Duong tron (C) c6 tam 1(1; 2), ban
kinh
R = 5 .
1) Tie'p tuyen di qua M va vuong goc voi IM nen nhan IM = (3;4) lam
VTPT
Nen phuong
trinh
tie'p tuye'n la: 3(x - 4) + 4(y - 6) = 0
<=>
3x + 4y - 36 = 0 .
2) Gpi A la tie'p tuye'n can tim.
Do A di qua N nen phuong
trinh
c6 dang
A:a(x + 6) + b(y-l) =
0<=>ax
+ by + 6a-b = 0, a^ + b^ (*)
Ta c6:
7a+ b
d(I,A)
= Ro
Va^ + b^
•
= 5o 7a + b
=
5^ o{7a + b)^

=25(3^
+b^)
o24a2+14ab-24b2
=0o24
- +
12 24
= 0c^
b
a = ^b
4
a=-lb'
3
3 3 7
• a=-b thay vao (*) ta c6: -bx + by + -b =
0o3x
+ 4y+ 14 = 0.
4 4
• a = —b thay vao n ta c6: —bx + by - 9b = 0 «• 4x - 3y + 27 = 0 .
3 3
Vay
CO
hai tie'p tuye'n thoa yeu cau bai toan la:
3x + 4y +14 = 0 va 4x - 3y + 27 = 0.
3) Goi
A(a;b).Tac6:
Ae(C)
(a-1)^
+(b-2)^ =25
<=>
a^ + b^

-2a-4b-20
= 0
a^ + b^
+5a-5b
= 0
lA.NA
= 0 [(a - l)(a + 6) + (b - 2)(b - 3) = 0
=^7a-b
+ 20 = 0
Tu
do ta suy ra duoc AeA:7x-y + 20 = 0.
Tuong tu ta cung c6 dug-c BeA=>AB = A=>AB:7x-y + 20 = 0.
2.
Cdch
lap
phimng
trinh
dizcrng
tron.
De lap phuong
trinh
duong tron (C) ta thuong su dung cac
each
sau
Cdch
7;Tim tam
I(a;b)
va ban
kinh
ciia duong tron. Khi do phuong

trinh
duong tron co dang: (x -a)^ +(y - b)^ = .
Cdch
2;Gia
su phuong
trinh
duong tron co dang: x^ + y^ - 2ax - 2by + c = 0
8
Cty
TNHH
MTV DWH
Khang
Viet
Dua vao gia thie't cua bai toan ta tim dugc
a,b,c.
Cach
nay ta thuong ap
dung
khi yeu cau viet phuong
trinh
duong tron di qua ba diem.
Vi
du
1.1.2.
Lap phuong
trinh
duong tron (C), bie't
1) (C) di qua
A(3;4)
va cac hinh chie'u ciia A len cac true toa do.

2 2 4
2) (C)
CO
tam nam tren duong tron (Cj): (x - 2)^ + y = - va tiep xiic voi hai
duong
thc^ng
A, :x-y = 0 va A2 :x-7y = 0.
Xffigidi.
1) Goi Ai, A2 Ian iugt la hinh chie'u ciia A len hai true Ox, Oy, suy ra
A,(3;0),
A2(0;4).
Giasii
(C):x^+y^-2ax-2by
+ e = 0.
Do
A,ApA2e(C)
nen ta co he:
-6a-8b
+ e = -25
-6a + c = -9
<=>
•!
-8b + e = -16
3
a = —
2
b
= 2.
e = 0
Vay phuong

trinh
(C): x^ + y^ - 3x - 4y = 0.
774
2) Goi
I(a;b)
la tam ciia duong tron (C), vi l€(Ci) nen: (a-2) +b =- (1)
Do (C) tie'p xuc voi hai duong thing A^Aj nen
d(I,
Aj) =
d(I,
A2)
a-b a-7b
<=>b = -2a,a = 2b
V2 5V2
• b = -2a thay vao (1) ta
CO
dugc:
(a - if- + 4a^ = - <=> 5a^ - 4a + — = 0 phuong
trinh
nay v6 nghiem
9 9 4 4 8 -^''i-'
• a = 2b thay vao(l)
taco:
(2b-2r+b''=-<::>b
= -,a =
o 00
Suy ra R = D(I,A,) =
Vay phuong
trinh
(C):

3. Cac
diem,
ctqc
biet
trong
tam
gidc.
Cho tam
giac
ABC. Khi do:
( 8l
2
r
4^
' 8
-:l.:J
x — +
y
I
5j
5 , 25
Phumig phdpgidi Todn Hiith hoc theo chiiyen de - Nguyen Phi't Klidnh, Nguyen Tat Thu
• Trong tam G
• True tam H :
3 ' 3
AH.BC
=
0
BH.AC
=

0
Tam duong tron ngoai tiep I:
lA^ = IB^
lA^ = IC^
• Tam duong tron noi tiep K :
Chu
y:C6 the tim K theo each sau:
* Gc
suy ra D
AB.AK AC.AK
AB AC
BC.BK BA.BK
BC AB
AB:
* Goi D la ehan duong phan giae trong goc A, ta c6: BD
=
DC , tu day
AC
AB;
* Ta CO AK = KD tu day ta c6 K.
BD ^
Tam duong tron bang tiep (goc A) J:
AB.AJ AC.AJ
AB AC
BJ.BC AB.BJ
BC AB
l?jdui.i.3.Cho tam giac ABC c6 A(1;3),B(-2;0),C
5 3
1) Tim toa do true tam H, tam duong tron ngoai tiep I va trong tam G cua
tam giac ABC. Tu dp suy ra I, G, H thang hang;

2) Tim toa do tam duong tron noi tiep va tam duong tron bang tiep goc A
cua tam giac ABC.
1) Taeo
Yc
1
Xffigidi.
1 9
3 8
Goi H(x;y), suy ra
AH
=
(x-l;y-3),BH
=
(x
+
2;y),BC
=
21 3
,AC =
( 3 _21
8' 8
in
CUj TNHH MTV DWH Khang Viet
Ma
<
AH.BC
=
0
nen ta eo
BH.AC

=
0
3 1
7(x-l) + (y-3)
=
0
j7x
+
y-10
=
0
(x
+
2)
+
7y
=
0 [x
+
7y
+
2 = 0
3
X = —
2
y = -:
1
2'
2
Suy ra H

Goi I(x;y), taeo:
x + y = l
21 3 111^
— x
+ —y
=
4 4^ 32
lA^ = IB^
IB^
= IC^
(x-l)2+(y-3f
=(x
+
2)^+y'
r
5^
2
]
=
x
o
+
y o
I
8y
V
8j
__15
16
31

15 31
16'16
Ta CO GH
=
^13 13^
, GI =
13.13
"l6'l6
>GH = -2GI. Suy ra I,G,H thang hang
[) Goi K{x; y) la tam duong tron noi tiep tam giac
ABC.
Ta c6:
KAB = KAC
KBC-KBA
<=>
AK,AB) = (AK,AC
= (BK,BC)
BK,BA
COS(AK,
AB)
= COS(AK, AC
cos (BK,
BA
j = COS ( BK, BC)
<=> <
AK.AB AK.AC AK.AB AK.AC
AK.AB AK.AC AB AC
BK.BA BK.BC BK.BA BK.BC
iBK.AB ' BK.BC I AB BC
Ma AK

=
(x-l;y-3),BK
=
(x
+
2;y),AB
=
(-3;-3)
nen (*) tuong duong voi
-3(x-l)-3(y-3) -8^^-^)-f^y-'^
37^
15N/2
3(x.2).3y 8^-"'^"^
2x - y = -1
x = 0
x-2y = -2 [y = l
3V2
I5V2
8 ^
Vay K(0;1).
Goi J(a;b) la tam duong tron bang tiep goc A eiia tam giac
ABC.
Ta co:
Phuvng
phlip
gidi
Todn
Hinh
hoc
theo

chiiyen
dc-
Nguyen
Phu
Khdnh,
Nguyen
Tat
Tltu
(ALAB) = (A1AC
(B],BC)
=
(BJ,AB
AJ.AB
_ AJ.AC
AC
AB
BJ.BC
_
BJ.AB
BC ~ AB
2a - b = -1
2a + b = -4
5
a = —
4
b
= -3-
2
Vay J
5,

4'
3^
'2.
4. Cdc
duang
ddc
hiet
trong
tam
gidc
4.1. Duang trung tuyen cua tam
giac:
Khi gap duong trung tuyen cua tam
giac,
ta chu yeu khai
thac
tinh
chat
di qua dinh va trung diem cua
canh
do'i dien.
4.2. Duong cao cua tam
giac:
Ta khai
thac
tinh
chat
di qua dinh va vuong
goc voi
canh

do'i dien.
4.3. Duong trung true cua tam
giac:
Ta khai
thac
tinh
chat
di qua trung
diem va vuong goc voi
canh
do.
4.4. Duong phan
giac
trong: Ta khai
thac
tinh
chat
ne'u M
thuoc
AB, M' doi
xung voi M qua phan
giac
trong goc A thi M'
thuoc
AC.
Vidu
7.i.4.Trong
mat ph^ng voi he tpa do Oxy, hay xac djnh toa do dinh C
cua tam
giac

ABC bie't rang hinh chie'u vuong goc cua C tren duong thang
AB la diem
H(-l;-l),
duong phan
giac
trong cua goc A c6 phuong
trinh
x-y + 2 = 0 va duong cao ke tuB c6 phuong
trinh
4x + 3y -1 = 0 .
JCffigidi
Ki
hi?u d, : X - y + 2 = 0, d2 : 4x + 3y -1 = 0.
Goi H' la diem doi xung voi H qua dj. Khi do H' E AC.
Goi A la duong thang di qua H va vuong goc voi dj.
x + y + 2=::0
Phuong
trinh
cua A:x + y + 2 = 0. Suy ra A n dj = I:
x-y+2=0
I(-2;0)
Nen ACndj = A:
Ta
CO
I la trung diem ciia HH' nen H'(-3;l).
Duong thang AC di qua H' va vuong goc voi dj nen c6 phuong
trinh
:
3x-4y
+ 13 = 0.

x-y+2=0
3x-4y
+ 13 = 0'
Vi
CH di qua H va vuong voi AH, suy ra phuong
trinh
cua CH:
3x + 4y + 7=0
[3x-4y
+ 13 = 0 3 4
•A(5;7).
12
Cty
TNHH
MTV DWH
Khang
Vie
Vi du
1.1.5.
Trong mat phang vai he toa do Oxy , cho tam
giac
ABC biet
A(5; 2). Phuong
trinh
duong trung true
canh
BC, duong trung tuyen CC
Ian lu^t la x + y- 6 = 0 va 2x-y + 3 = 0. Tim toa do cac dinh B,C cua tam
giac
ABC.

Xgfi
gidi.
Goi d:x + y-6 = 0, CC: 2x-y + 3 = 0 . Ta c6:
C(c;2c
+ 3)
Phuong
trinh
BC :x-y + c + 3 = 0
Goi M la trung diem ciia BC, suy ra M:
3-c
x + y-6 = 0
x-y+c+3=0
X =
-
y=-
2
c + 9
Suy ra
B(3-2c;6-c)=>C'(4-c;4-|)
Ma
C'eCC
nen ta c6:
2(4-c)-(4 )
+ 3 =
0<=> c
+7 = 0^ c = — .
2 2 3
Vay B
19.4
3 '3

, C
14 37
3' 3
5. Mot
sobdi
todn
dung
hinh
ca ban.
5.1.
Hinh
chie'u vuong goc H cua diem A len duong thang A
• Lap duong thang d di qua A va vuong goc voi A
• H=dnA
5.2. Dung A' doi xung voi A qua duong thang A
• Dung hinh chie'u vuong goc H cua A len A
Lay A' do'i xung voi A qua H:
'^A'-^Xj^
x^
lyA'=2yH-yA
5.3. Dung duong tron (C) do'i xung voi (C) (c6 tam I, ban
kinh
R) qua duong
thSng
A
• Dung r doi xung voi I qua duong thang A
• Duong tron (C) c6 tam
I',
ban
kinh

R.
Chii
y:
Giao
diem ciia (C) va (C) chinh la
giao
diem cua va A .
5.4. Dung duong thang d' doi xung voi d qua duong thang A .
• Lay hai diem M,N
thuoc
d. Dung M',N' Ian luot doi xung voi M, N qua A
'if!', r<(.:
• d' = M'N'.
Phumig
phdp gidi
Todii
Uinh
hoc theo chuyen dc - Nguyen
Pliii
Khdnh,
Nguyen Tat
Thti
Vidu
1.1.6.Trong mat phang Oxy cho duong thang
d:x-2y-3
= 0 va hai
diem A(3;2), B(-l;4).
1) Tim diem M thuoc duang thang d sao cho MA + MB nho nhat,
2) Viet phuong
trinh

duang thang d' sao cho duong thang A: 3x + 4y +1 = 0
la
duong phan
giac
ciia goc tao boi hai duong thang d va d'.
JCffigidi.
1) Ta tha'y A va B nam ve mot phia so voi duong thang d. Goi A' la diem doi
xung
voi A qua d. Khi do vai moi diem M thuoc d, ta luon c6: MA = MA'
Dodo:
MA + MB = A'M + MB>A'B.
Dang thuc xay ra khi va chi khi M = A' B n d.
Vi
A'A 1 d nen AA' c6 phuong
trinh:
2x + y -8 = 0
19
2x+y-8=0
Goi H = dnAA'=>H:<^
x-2y-3=0
Vi
H la
trung
diem ciia AA' nen
23
'23 _6
5' 5
•A'
yA'
= 2yH-yA=-5

Suy ra A'B =
28 26
5 5^
trinh
A'B :]3x +
14y-43
= 0
, do do phuong
Nen M:
x-2y-3=0
13x +
14y-43
= 0
<=> <
X
=
-
16
5 ,
J_
10
•M
16 J_
5 '10
2) Xet he phuong
trinh
x-2y-3 = 0 rx = l
<=><^ , suy ra dnA =
I(l;-l)
3x + 4y + l = 0 [y l

Vi
A la phan
giac
cua goc hgp bai
giiia
hai duang thang d va d' nen d va
d' do'i xung nhau qua A , do do led'.
'3 _16'
.5'"5
Lay
E(3;0)
G
d, ta tim
dugc
F
la
diem do'i
xiing
vai E qua A, ta c6
Fed'. Suy ra
FI
=
(2 U
5'5
, do do phuong
trinh
d': llx - 2y -13 = 0 .
14
Cty
TNHH

MTV DWH
Khang
Viet
CP BAI
TAP
Bai
l-l-l-
Trong mat phang Oxy cho tam
giac
ABC
CO
A(2;l), B(4;3),
C(-3;-l)
1) Tim toa do true tam, tam duong
tron
ngoai tiep tam
giac
ABC
2) Viet phuong
trinh
duong
tron
ngoai tiep tam
giac
ABC.
Jiuang
ddn gidi
1) Goi H(x;y) la true tam tam
giac
ABC, ta c6:

AH.BC = 0
BH.AC = 0
'(x -
2)(-7)
+ (y
-1)(-4)
= 0 J7x + 4y -18 = 0
(x -
4)(-5)
+ (y -
3)(-2)
= 0 ^ [Sx + 2y - 26 = 0 ^
X
=
34
y = -
46
Vay H
34 46
Goi I(x;y) la tam duang
tron
ngoai tiep tam
giac
ABC, ta c6:
•(X
- 2)2 + (y -1)2 = (X - 4)2 + (y - 3)2 fx + y = 5
IA2
= IB2
IA2
= IC2

(x-2)2+(y-l)2
=(x + 3)2+(y + l)2 [8x + 4y =-5
x = —
25
y =
-
45
Vay I
25_45
4 ' 4
2) Duong
tron
ngoai tiep tam
giac
ABC c6 ban
kinh
R = lA =
Nen no phuong
trinh
la:
V2770
(
25^
2
45^
^ 1385
x + —
+ y-
V
—

I 4,
y-
V
4y
8
Bai
1.1.2.
Trong mat phang toa do Oxy cho tam
giac
ABC c6 A(3;2) va
phuong
trinh
hai duang
trung
tuyen BM: 3x + 4y - 3 = 0,CN: 3x - lOy -17 = 0.
Tinh
toa do cac diem B, C.
Jiuang
dan gidi : ? ; • ;
Goi G la trong tam ciia tam
giac,
suy ra toa do ciia G la nghiem cua he
'3x + 4y - 3 = 0
3x-10y-17 = 0
7
^ = 3
[y
=
-l
>

; r J J' .I'i-
Phumig
phdpgiiii
Toan Hitih hoc
theo
chuyen
de-
Nguyen
Phi'i
Khanh,
Nguyen
Tat Thu
Goi
E la
trung diem ciia
BC, suy ra EA = -GA
=>
E(2; .
Gia
sir B(a;b), suy ra C(4-a;-5-b). Tu do ta c6 h^:
3a + 4b-3 = 0
a = 5
b
= -3'
3a + 4b - 3 = 0
"
3(4-a)-10(-5-b)-17 = 0 [-3a + 10b + 45 = 0
Vay
B(5;-3),C(-l;-2).
Bai

1.1.3.
Trong
mat
phang
toa do Oxy cho tam giac ABC c6 A(-3;0) va
phuong
trinh
hai
duong phan
giac
trong
BD: x - y -1 = 0,CE : x + 2y +17 = 0 .
Tinh
toa do cac
diem
B, C.
Jiu&ng ddn gidi
Gpi
A^ doi
xiing
voi A qua BD, suy ra Aj e BC va
A^(l;-4)
Aj
do'i
xung
voi A qua CE, suy ra A2 e BC va
A2(-—;-—).
5 5
Suy ra
phuong

trinh
BC : 3x - 4y -19 = 0 .
x-y-l=0
Toa dp B la
nghi^m
cua he:
Toa do C la
nghiem
cua he:
x = -15
3x-4y-19 = 0 [y = -16
x + 2y + 17 = 0 fx = -3
B(-15;-16).
•C(-3;-7).
3x-4y-19 = 0 [y = -7
Vay
B(-15;-16),C(-3;-7).
Bai
1.1.4.Trong
mat
phSng
toa do Oxy cho tam giac ABC c6 C(5;-3) va
phuong
trinh
duong
cao
AA':x-y
+ 2 = 0,
duong trung tuyen
BM:

2x + 5y -13 = 0.
Tinh
toa do cac
diem
A, B.
Jiixang ddn gidi
Ta
CO
phuong
trinh
BC: x + y - 2 = 0
fx
=
-l
Suy ra toa do
ciia
B la
nghiem
cua he:
x+y-2=0
2x + 5y-13 = 0 ly = 3
•B(-l;3).
Gpi
A(a;a + 2), suy ra toa do
ciia trung diem
AC la M
+
5 a-1^
Ma MeBM
nen 2^y^ + 5^-13 = 0«a = 3 =^

A(3;5).
Vay
A(3;5),B(-1;3).
Bai
1.1.5.
Trong
mat
phang
toa dp Oxy cho tam giac ABC CO
B(l;
—3) va
phuong
trinh
duong
cao
AD:2x-y
+1 = 0,
duong phan
giac CE:x + y- 2=::0
.Tinh
toa dp cac
diem
A, C.
Cty TNHH MTV DWH Khang Viet
Jiic&ng ddn gidi
Ta
CO
phuong
trinh
BC:x + 2y + 5 = 0.

[x
+ y-2 = 0 [x = 9
Tpa
dp
diem
C la
nghiem
'^"^ L ^ 2y + 5 = 0 ^ |y = -7
Gpi
B' la
diem
doi
xung
voi B qua CE, suy ra B'(5;l) va B' e AC
Do
do, ta
CO
phuong
trinh
AC :2x + y- ll = 0.
5
C(9;-7).
Toa dp
diem
A la
nghiem ciia
he:
2x-y+l=0
2x +
y-ll

= 0
2 => A
y
= 6
2
Vay
A
5;6
2
,C(9;-7).
A(l;2)
Bai
1.1.6.
Trong
mat
phang
voi h^ tpa dp Oxy, cho tam giac ABC co M (2; 0)
la trung diem
cua canh AB.
Duong trung tuyen
va
duong
cao qua
dinh
A Ian
lupt
CO phuong
trinh
la 7x - 2y - 3 = 0 va 6x - y - 4 = 0 .
Viet phuong

trinh
duong thang
AC.
Jiu&ng ddn gidi
|'7x-2y-3 = 0
Toa do A
thoa
man he:
<^
•
•
[6x-y-4
= 0
Vi
B
do'i xiing
voi A qua M nen suy ra B = (3; - 2).
Duong
thSng
BC di qua B va
vuong
goc voi
duong
thSng:
6x - y - 4 = 0 nen
suy
ra
Phuong
trinh
BC: x + 6y + 9 = 0 .

'7x-2y-3 = 0
'x+6y+9=0
Tpa
dp
trung diem
N cua BC
thoa
man he:
•N
Suy ra AC =
2.MN
= (-4; - 3).
Phuong
trinh
duong th^ng
AC : 3x - 4y + 5 0.
^
;
Bai
1.1.7.
Trong
mat
phSng
Oxy cho
duong tron
(C): (x - if + (y -1)^ = 25 .
1) Lap
phuong
trinh
tiep tuyen

cua (C),
biet tiep tuyen
di qua A(3;-6)
2) Tu
diem
D(-4;5) ve de'n (C) hai
tiep tuyen DM,
DN (M, N la
tiep diem). Viet
phuong
trinh
duong thang MN.
,, (.
J^lurnigd&ngidi ,ob
«.M
(
+
Duong tron (Qxd-taDL.K2i
1), ban
kinh
R = 5 . Av isH
THU
ViEN
Tl.VHBtNH
THU.AN]
1
Phumig
phtip
giai
Toan

Hinh
hoc
theo
chuyen
dS"-
Nguyen
Phu
Khdnh,
Nguyen
Tat
Thii
1) Gia six A : ax + by + c = 0 la tiep tuyen ciia (C)
Do B e A nen 3a - 6b + c = 0 => c = 6b - 3a
2a + b + c
A
la tiep tuyen ciia (C) nen
d(I,
A)
= R
Va^+b^
=
5<=>
-a + 7b
= 5
ol2a^
+7ab-12b^
= 0«
4
a=-ib
3

Tu
do, ta
CO
dugc phuong
trinh
tiep tuyen la:
3x + 4y +15 = 0 va 4x - 3y - 30 = 0 .
fTe(C)
2) Goi T(X(,;yQ) la tiep diem , ta c6:
DI.IT
= 0
<=>
(Xo-2)2+(y„-l)2=25
(xo-4)(xo-2)
+
(yo4-5)(yo-l)
= 0
Xo+yo-4xo-2yo=20 _ , ^
^2xo-6yo-23
= 0
Xo+yo-6xo+4yo=-3
Vay phuong
trinh
MN : 2x - 6y - 23 = 0 .
§ 2. XAC
DINH
TOA DO CUA MQT
DIEM
Bai toan co ban ciia phuong phap toa do trong mat phang la bai toan xac
dinh

toa do ciia mot diem.
ChSng
han, de lap phuong
trinh
duong thang can
tim
mot diem di qua va VTPT, voi phuong
trinh
duong tron thi ta can xac djnh
tarn
va ban kinh Chung ta co the gap bai toan tim toa do ciia diem dugc hoi
true tiep
hoac
gian tiep.
• Ve phuong dien hinh hgc tong hgp thi de xac
dinh
toa do mot diem, ta
thuong chiing minh diem do thugc hai hinh (H) va (H'). Khi do diem can tim
chinh la
giao
diem ciia (H) va (H').
• Ve phuong di^n dai so, de xac
dinh
toa do ciia mot diem (gom hai toa do) la
bai toan di tim hai an. Do do, chiing ta can xac djnh dugc hai phuong
trinh
chiia hai an va giai he phuong
trinh
nay ta tim dugc toa do diem can tim. Khi
thiet lap phuong

trinh
chiing ta can luu y:
+) Tich v6 huong ciia hai vec to cho ta mgt phuong
trinh,
+) Hai doan thang bang nhau cho ta mgt phuong
trinh,
+) Hai vec to bang nhau cho ta hai phuong
trinh,
18 '
Cty TNHH MTV DWH
Khang
Viet
+) Neu diem M e
A
: ax + by + c = 0,a ^ 0 thi M
-bm-c
-;m
,
liic
nay toa
do ciia M chi con mgt an va ta chi can tim mgt phuong
trinh.
Vi
da 1.2A. Trong mat phang Oxy cho duong tron (C): (x
-1)^
+ (y
-1)^
= 4
va duong thang A:x-3y-6 = 0. Tim tga dg diem M nam tren A , sao cho
tvr

M ve dugc hai tiep tuyen
MA,
MB (A,B la tiep diem) thoa
AABM
la tam
giac
vuong.
Xgigiai
Duong tron (C) co tam
1(1;
1), ban
kinh
R = 2 .
Vi
AAMB vuong va IM la duong
phan
giac
ciia goc AMB nen AMI = 45°
Trong tam
giac
vuong lAM , ta co:
IM
= 2V2, suy ra M thugc duong
tron
tam I ban
kinh
R' = 2 .
Mat
khac
Me A nen M la

giao
diem ciia A va (I,R'). Suy ra tga do
ciia M la nghiem ciia he
x=3y+6
x-3y-6=0
(x-i)2
+(y-i)2 =8 " [(3y +
5)2
+(y-l)' =8
'x = 3y + 6
5y^ +14y + 9 = 0
y
= -l,X = 3
= =-
^~ 5'^ 5
(3 9 I
Vay
CO
hai diem Mj (3; -l) va M2 -; — thoa yeu cau bai toan.
Vi
du
1.2.2.
Trong mat
phSng
voi he tga do Oxy cho cac duong thang
di:x
+ y + 3 = 0, dj :x-y-4 = 0, dg :x-2y = 0. Tim tga do diem M nam
tren
duong
thSng

sao cho khoang
each
tu M den duong thang
d^
bang
hai Ian khoang
each
tu M den duong thang d2 .
Xffi
gidi
3y + 3
-4
Taco
Med3,suyra M(2y;y). Suy ra d(M,di) =
—^;d(M,d2)
= ^^
Theo
gia thiet ta co: d(M,di) =
2d(M,d2)
<^
3y + 3 ^2ly-4
Phuvng
plidp
gidi
Toiin
Hiith
hoc
theo
chuyen
dc- Nguyen

Phii
Klidnh,
Nguyen Tn't Thu
<=>
3y+3=2y-8
3y + 3 = -2y +
• Voi y =
-n^M(-22;-ll).
• Voi y = l^M(2;l).
<=> y = -ll;y =
1
.
Vi du 1.2.3. Trong he toa do Oxy, cho diem A(0; 2) va duong th3ng
d:x-2y + 2 = 0. Tim tren duong thang d hai diem B, C sao cho tam
giac
ABC vuong 6 B va AB = 2BC .
JCffigidi
Ta
CO
AB 1 d nen AB c6 phuong
trinh
: 2x + y - 2 = 0.
x-2y+2=0
TQa dp diem B la nghiem ciia he :
<|
^ ^ \
„ 2^5 ^„ AB S
Suy ra AB = => BC = -— = —.
^ 5 2 5
Phuong

trinh
duong
tron
tarn
B, ban
kinh
BC = — la:
5
'2.6'
5'5
( 2^
2
6^
2
1
X
—
+
y-
V
— - —
I
5,
y-
V
5y
5
x-2y + 2 = 0
Vay toa dp diem C la nghiem ciia he :
Vay

CO
hai bp diem thoa yeu cau bai toan la:
( 2^
f
6l
X
—
+
y"
I
5,
v
5j
5
x=ay=l
4 7
B
'2 6^
5'5
,
C(0;l)
va B
2 6
5'5
, C
4 7
Vi du
1.2.4.
Trong mat phang voi h^ tpa dp Oxy, cho diem A(2; 2) va hai
duong thSng: di:)^ + y-2=^0, d2:x + y-8 = 0. Tim tpa dp diem B, C Ian

lupt
thupc di, d2sao cho tam
giac
ABC vuong tai A.
Xffigidi
Vi
B6di=^B(b;2-b);Ced2=^C(c;c-8).
Xi.AC
= 0
f(b-l)(c-4)
= 2
Theo
de bai ta c6 he:
AB = AC
Dat x = b-l;y = c- 4 taco:
Vay
B(3;-1);C(5;3)
hoac
B(-1;3),C(3;5)
.
(b-lf-(c-4f .3
xy = 2
x
= 2
x
-
-2
.x^-y^
<=> •
V <

.x^-y^
= 3
.y
= i
.y
=
-1
Cty
TNHH MTV DWH
Khang
Viet
Vi du
1.2.5.
Cho parabol (P): y^ = x va hai diem A(9; 3), B(l; -1) thupc (P).
Gpi
M la diem thupc cung AB cua (P) (phan ciia (P) bi chan boi day AB).
Xac
djnh
tpa dp diem M nam tren cung AB sao cho tam
giac
MAB c6 dien
tich
ion nha't.
JCgi
gidi.
Phuong
trinh
AB : x - 2y - 3 = 0
Vi
M

G
(P) => M(t^; t) tu gia thiet suy ra -1 < t < 3
Tam
giac
MAB c6 dien
tich
ion nha't o d(M, AB) Ion nha't
t^
-2t-3
-,te(-];3). Ma
d(M;AB)
=
Suy ra maxd(M, AB) = dat dupe khi t =
11=>
M(l;l).
v5
Vi du .2.6. Trong mat phang Oxy cho duong
tron
(C): (x -1)^ + y^ = 2 va
hai
diem A(l;-1), B(2;2). Tim tpa diem M thupc duong
tron
(C) sao cho
dien
tich
tam
giac
MAB bang ^ .
Xffi
gidi.

Ta
CO
AB = Vio va
S^^^AB
=
-d(M,AB).AB
=
d(M,AB)
=
Lai
CO
AB = (1;3) nen n =
(3;-l)
la
VTPT
ciia duong thang AB
Suy ra phuong
trinh
AB : 3(x
-1)-(y
+1) = 0 hay 3x-y-4 = 0.
Gpi
M(a; b) e (C) => (a - if + b^ = 2
, 3a-b-4
Khi
do d(M; AB) - -=
<=>
;=— - ^
VIO vio VlO
Ta

CO
he phuong
trinh:
(a-l)2+b2=2 f(a-l)2 + b2=2 , .
<=i>r
hoac
3a-b-4 =1
hoac
3a - b - 4 = 1
(a-l)2 + b^ =2
b
= 3a - 5
(a-1)2+(3a-5)2 =2
b
= 3a - 5
3a-b-4 = l
(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
21
Phuang

phdp gidi Todn
Hinh
hgc theo chuyen
de-
Nguyen Phu
Khdnh,
Nguyen Tat Thu
5a^ -16a+ 12 = 0
hoac
b
= 3a-5
<=>
i
_12
_4
^"T'''"
5 hoac
b
= 3a-5
5a^ -10a+ 4=0
b
= 3a-3
5±V5
a =
•
b
= 3a-3
Vay
CO
boh diem thoa dieu kien bai toan la:

12
m r4 13^ _
fS-VS
-375
Ml
va
M4
5 + V5
375
5
5
m BAI
TAP
Bai
1.2.1.
Trong
mat
phang
voi
he toa do
Oxy,
cho
duong tron (C)
c6
phuong
trinh:
(x -1)^ + y^ = 1. Ggi
I la
tarn ciia (C). Xac dinh toa
do

diem
M
thuQC (C) sao cho IMO = 30".
Jiudng ddn
gidi
Ggi
diem M(a;b).Do Me(C)
nen {a-lf
+h^=l.
Mat
khac
O € (C)
=:>
lO =
IM
= 1.
Tam
giac
IMO
c6
OIM -120°
nen OM^ = lO^ +
IM^
-
2IO.IM.cos
120°
o
a^ + b^ =
3.
Toa do diem M la nghiem cua he

:
(a-l)'+b2=l
a^+b^
=3
3
a =
—
2
b
= +
Vay
M =
Til
^2
Bai 1.2.2.
Trong
mat
ph^ng voi
he toa do
Decac
vuong
goc
Oxy,
cho
parabol (P) c6 phuong
trinh
y^ = x
va
diem 1(0; 2). Tim toa
do

hai diem M,
N
thuQC (P) sao cho
IM
= 4iN
.
Jiuong dan
gidi
Vi
M,Ne(P)
nentaco
M(m^;m),N(n^n).
SuyraIM=
m'^;m-2,IN=
n^n-2
.Dodo:
22
Cty
TNHH
MTV DWH Khang Viet
IM
=
4IN
<::>
m^ =4n2
m
-
2
= 4(n
-

2)
<=>
s
m
=4n-2
(2n-l)2
=n2
n=l
m
= 2
V
{
Vay
CO
hai cap diem thoa yeu cau bai toan la:
1
l^
M(4;2),N(1;1)
hoac
M
9'
3J
9'3
1
n
=
—
3
2 •
m

=
—
3
Bai 1.2.3.
Trong mat
phang
toa dp Oxy cho diem
A(3;2),
cac duong thang
dj
:
X
+ y
-
3 = 0
va:
d2
:
x + y
-
9 = 0 . Tim
toa do
diem B
G
dj,
va
C
e
d2
sao

cho tam
giac
ABC vuong can tai A.
,
Jiuong<Mngidi
r ,s'r.
Vi
Bedj :x + y-3 = 0
nen
B(b;3-b), Cedj :x + y-9 = 0
nen
C(c;9-c).
De tam
giac
ABC vuong can tai A khi va chi khi
Hay
Dat
u
= b
-
3,
v = c
-
2, ta c6:
AB
=
AC AB2=AC2
<=>
i
AB

1
AC
AB.AC
= 0
(b-3)2+(b-l)2=(c-3)2+(c-7)2
(b-3)(c-3) + (l-b)(7-c) = 0
u^+(u
+ 2)^
=(v-l)2
+(v-5)^
u(v-l)
+ (u + 2)(v-5) = 0
3u
+
5
<=>
(u
+
1)^
=(v-3)^
+3
uv-3u+v-5=0
3u
+
5
,
v
= u =
l
u

+
1
<=><^
V •!
x2
.
I
V
=
4
v
=
•
u
+ 1
•
+
3
u
=
-3
v
= 2
(u
+
1)^
=4
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).
Chti
i/ Ngoai
each
tren, ta c6 the giai
theo
each
khac
nhu sau:
Tjnh
tien
he
true toa dp Oxy ve he tuc
XAY
theo
vec to OA, ta c6 cong
thuc
'x
=
X
+
3
' ;
J
= Y + 2
=

Trong
he true moi,
ta
c6 phuong
trinh
cua dj
:X
+
Y
+
2 = 0, d2 :X + Y-4 = 0.
Vi
tam
giac
ABC vuong can tai A nen
phep
quay
Q
Q : B -> C
(A,±90
) |,
,
, , '23
doi
true:
Phumtg
phapgiiii
Toan
Hinh
hqc

theo
chuyen
rfe -
Nguyen
Pht'i
Khanh,
Nguyen
Tat Thu
Ma B e => C e d
1
= Q^^.^^^jo/cli), do do C ^ d2 n d,.
• Xet phep quay Q^^
^^^^^,
ta c6 phuong
trinh
d, : X - Y - 2 = 0
Do do toa dp cua C la nghiem cua he:
X-Y-2=0
X+Y-4=0
<=> {
X = 3
Y=r
Xet phep quay Q^^ ^^^(y ta c6 phuong
trinh
d j : X - Y + 2 = 0
Do do tga do cua C la nghiem cua he:
•
^ ^ ^ ^ ^ <=>
^ • •
X+Y-4=0

X = l
Y = 3'
x = 6
y
= 3
x = 4
y
= 5-
Tu
do ta tim dugc B, C.
Bai
1.2.4.
Trong he true toa do Oxy cho AABC voi A(2;3), B(2;l),
C(6;3),
Goi
D la giao diem cua duong phan
giac
trong goc BAG voi BG. Tim tat ca cac
diem
M thuoc duong tron (G): (x - 3)^ + (y -1)^ = 25 sao cho : 5^^^ =
2SADB
•
Jiuang
ddn
gidi
Ta CO AB = (0; 2), AC = (4; 0), BG - (4; 2)
DG AG 2 BG 3 3 ^3 3^ I 3 3^
Phuong
trinh
AB: x - 2 = 0,

nen
d(D,
AB)
=
|
=^
S^^BD
=
{AB.d(D,AB)
=
1-2.^
=
1
Phuong
trinh
DG : x - 2y = 0 . Goi M(a; b) => (a - 3)^ + (b -1)^ = 25 (1)
Mat
khac:
'AMCD
= 2S
-GD.d(M,GD)
= -«i ix/s.
2 ^323
AABD
a-2b
8
=
—
<=>
3

a-2b =4
o a = 2b + 4
hoac
a = 2b - 4
a = 2b - 4 thay vao (1) ta c6 duoc: (2b - 7)^ + (b -1)^ = 25
<=>
b^ - 6b + 5 = 0
"b = l=^a = -2=>M(-2;l)
b
=
5=>a
= 6=>M(6;5)
a = 2b + 4 thay vao (1) ta c6 dugc: (2b +1)^ + (b -1)^ = 25 <r> 5b^ + 2b - 23 = 0
b
=
=e>a
= ^—=>M
.
-1-2729
18-4729
b
=:
=> a =
.M
-1
+
2729
18 +
4729
5 . ' 5

y
'-1-2729
18-4729^
24
Cty
TNHH
MTV DWH
Khang
Vie
I
Bai
1.2.5.
Trong mat phSng voi he true toa do Oxy, cho duong thSng
j.x-3y-4 = 0 va duong tron (C): x^ + y^ - 4y = 0 . Tim M thugc d va N
thupc (C) sao cho chung doi xung qua A(3;l).
1,,;f.„f;
Jiuang
ddn
gidi
Vi
M e d M(3m + 4; m). Do N doi xung voi M qua A nen N(2 - 3m; 2 - m)
MaNe(G) T.r ^
nen (2 - 3m)^ + (2 - m)^ - 4(2 - m) = 0 o lOm^ - 12m - 0 o m = 0,m = -
Vay
CO
hai cap diem thoa yeu cau bai toan:
M(4;0),N(2;2) va M
386
I
5 '5j

' 8 4^
•5'5
Bai
1.2.6.
Trong mat phSng Oxy cho diem A(l;4). Tim hai diem M,N Ian lugt
nam tren hai duong tron (q):(x-2)2+(y-5)2 =13 va {C^):{x-lf+{y-2f ^75
sao cho tam
giac
MAN vuong can tai A.
Jiuung
ddn
gidi
Xet phep quay
Q^^.^^^,0)
: M N va (Gj) ^ (Gj)
Ma Me(G,)^N€(Gj)^N€(G2)n(Gi).
• Voi Q^^
^^(y
ta
CO
phuong
trinh
(Gj) :x^+(y-5)^ -13
Toa dp diem N la nghiem cua he:
x2+(y-5)^=13
(x-i)2+(y_2)2=25
x2 +y2 _i0y + 12 = 0
x2 +
y2-2x-4y-20
= 0

x2+y2-10y +
12=:0^
J5y2 - 53y +134 = 0
x = 3y-16
-1
+
37T29
10
53 + 7129
10
X -
y
=
-
V
<
X =
-
x = 3y-16
-1-37T29
10
y
=
-
53-7129
10
Truong
hop nay c6 hai bp diem:
M
Va M

23 +
7T29
_
51 -
37T29
10 ' 10
'23 - 7129 51 +
37l29
f-l
+
37l29
53 +
7l29
10
10
10
10
-l-37l29
53-7l29
10
10
Phumtg phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phti Khihih, Nguyen Tat Thu
Q(A-9oO)'*^™P''^""^*""^
(Ci):(x-2)2+(y-3)2 =13
Toa do diem N la nghiem ciia he:
(x-2)2+(y-3)2
=13 {x
=
A \x
=

5
<=> < V <
(x-])2+(y-2)2 =25 ly = 6 [y = 5
Tmonghgp nay c6 hai bo diem: M(-1;7),N(4;6) va M(0;8),N(5;5).
Bai 1.2.7. Trong mat phSng Oxy cho duong tron (C): (x - if + (y - 4)^ = y
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 ll-5c
. Do G nam tren duong tron (C) nen ta c6
phuong trinh: i^^ + i^^lHL = ^ <:>29c2
+
114c4-85
= 0
<^ c =-l,c =
9 36 9
29
Vay
CO
hai diem C thoa yeu cau bai toan la: Cj (-1;8), C-
85.372

29'
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°
Do do: MI = ^
=
iS IM^ = 20
sin30"
Do Med nen suy ra
M(XQ;
XQ +1)
Khi do ta c6: MI^ = (X(,
+1)^
+ (x„ -1)^ = 20 o x^ = 9 x^ = 3; XQ = -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
ABC
banglS.
26
Cty TNHH MTV DWH Khang Vi?t
Jiuang ddn gidi
Khi do di?n tich tam giac ABC la: S^BC = ^
AB.d(C,
A)
= 3AB
Theo gia thiet ta c6:
AB
=
5

<^ (4 - 2a)^ +
Vay hai diem can tim la A(0;1) va B(4;4).
r6-3a^
2
= 25c*
"a = 4
I
2 ,
a = 0
2 .2
va
' X y
Bai 1.2.10. Trong mat phang voi he toa dp Oxy cho elip (E): — + — = 1
hai diem
A(3;-2),
B(-3;2).
Tim tren
(E)
diem C c6 hoanh dp va tung dp
duong sao cho tam giac
ABC
c6 dien tich Ion nha't.
>
j i
Jiuang ddn gidi <
Ta
CO
phuong trinh duong thang
AB:
2x

+
3y = 0 ,
Gpi
C
(x; y) voi x > 0, y > 0. Khi do ta c6 + =
^
dien tich tam giac
ABC
la SABC =
2^^-^(C'^^)
= ^|2^ + M = 3^^
<3
3 4
'85
13'
Dau bang xay ra khi
9 4
3 2
x = 3
2 . Vay C
2 2^
9 4
(
3V2
= 3
1170
13
§ 3. NHOM CAC BAI TOAN VE 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
jyong 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
Xgi
gidi.
Vi
Aed,,Ced2 nen A(2a -
l;a),C(3c;-2c),
suy ra I

r2a + 3c-l a-2c
la
trung
diem AC
Do ABCD la hinh vuong nen 1 la trung diem cua BD, hay I e Ox .
Do do a = 2c .
Mat
khac
AC 1 BD = Ox nen suy ra 2a -1 = 3c o c = 1.
Tu
do, ta tim
dugc
A(3;2),
C(3;-2),
1(3; 0).
Vi
B e Ox =^
B(b;0),
ma IB = IA = 2 =^ |b -
3|
= 2 o b = 5,b = 1.
Vay toa do cac dinh ciia hinh vuong ABCD 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).
Vidu
7.3.2.Trong
mat phang Oxy cho ba diem 1(1; 1),
J(-2;2),
K(2;-2).
Tim
toa do cac dinh cua hinh vuong ABCD sao cho I la tam hinh vuong, J
thuoc
canh
AB va K
thuoc
canh
CD.
Goi J' doi xiing vai J qua I, ta c6
J'(4;0)
va J' € CD .
Ta c6: KJ' = (2; 2), suy ra phuong
trinh
CD:x-y-4 = 0.
Vi
AB//CD nen phuong
trinh
AB:x-y+4=0.
Do d(I,AB) = 272 nen suy ra
AB = 4V2 => IA = 4
A
e AB => A(a; 4 + a), do do
IA
=

4o(a-l)^
+(a + 3)^
=16«.a^
+ 2a-3 =
0<=>a
= l,a = -3
• a =1, ta
CO
A(l;3),
B(-3;l),
C(l;-1), D(5;l)
• a = -3, ta
CO
A(-3;l),
B(l;3),
C(5;l), D(l;-1).
Vidu
2.3.3.Trong
mat phang Oxy cho duong tron (C): (x - 2)^ + (y - 1)^ = 10.
Tim
toa do cac dinh cua hinh vuong MNPQ, biet M trung voi tam cua
duong tron (C); hai dinh N, Q
thuoc
duong tron (C); duong thang PQ di qua
E(-3;6) va
XQ
> 0 .
Ta
CO
M(2;l) va EQ la tiep tuyen cua (C).

Phuong
trinh
EQ c6 dang:
28
Cty TNHH MTV DWH
Khang
Viet
a(x + 3) + b(y - 6) = 0 o ax + by + 3a - 6b = 0
Vi
d(M,EQ) = 7i0 ^ E
5a-5b
nen ta co:
= >/To
ci>(5a-5b)^
=10(a2+b2)
<:i>3a^
-10ab + 3b^ = 0
<=>
a = 3b,b = 3a
, a = 3b, ta c6 phuong
trinh
EQ: 3x + y + 3 = 0.
Khi
do toa do Q la nghiem ciia he
(x-2)2+(y-l)2=10
3x + y + 3 = 0
x = -l
y
= 0
Truong hop nay ta loai vi

XQ
> 0 .
• b = 3a, ta
CO
phuong
trinh
EQ: x + 3y -15 = 0 . Khi do toa do Q la nghiem
fx = 3
Ciia
he
(x-2)2+(y-l)2=10
3x + y + 3 = 0
<=> <
y
= 4
•Q(3;4).
Taco
P(15-3x;x)
va QP = MQ => (12 - 3x)^ + (4 - xf = 10 x = 3,x
X
= 3, ta
CO
P(6;3),
suy ra tam cua hinh vuong
1(4;2)
nen
N(5;0)
X
= 5, ta
CO

P(0;5),
suy ra tam cua hinh vuong
1(1;3)
nen N(-l;2).
Vay
CO
hai bo diem
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).
= 5
Vi
du
1.3.4.
Trong mat phang voi he toa do Oxy cho hinh chir nhat ABCD
CO
diem 1(6; 2) la
giao
diem cua 2 duong
cheo
AC va BD. Diem M(1; 5)
thuQc duong th3ng AB va trung diem E ciia
canh
CD
thuoc
duong thang
d:x + y- 5 = 0. Viet phuong
trinh

duong thang AB.
gidi.
Vi
E€d=:>E(a;5-a)=>iE
=
(a-6;3-a).
Goi N la trung diem cua AB, suy ra I la trung diem cua EN nen :
X^
N:
=2xi
-Xg
=12-a
XN
=2y,-yE
=a-l
•N(12-a;a-l)
•MN-(ll-a;a-6).
Vi
E1MN => MN.IE = 0
<»(ll-a)(a-6)
+
(a-6)(3-a)
=
0<r>
a = 6
a = 7'
Phumtgphdp gidi Todn Hinh hoc
theo
chuyen de-
Nguyen

Phu Khdnh,
Nguyen
Tat Thu
a = 6 => MN =
(5;0),
suy ra phuang
trinh
AB:y-5 = 0
• a = 7 => MN =
(4;1),
suy ra phuong
trinh
AB: 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
di:x-y-3 = 0 va
d2:x
+ y- 6 = 0. Trung diem cua AB la
giao
diem aia
dj voi true Ox. Tim toa dp cac dinh cua hinh chi> nhat.
Taco
dj ndj =1:
x-y-3=0
(9 3^

x + y-6 = 0 yi 1,
Goi M la
giao
cua duong thang dj voi Ox, suy ra M(3;0).
Vi
AB 1 MI nen suy ra phuong
trinh
AB:x + y- 3 = 0
c
AD = 2MI = 3 V2 =^ AB == 2 72 =^ AM = 2
AD
Ma A€ AB=^
A(a;3-a)i=>
AM^ =
2o(a-3)^
=1 <:i>a =
2,a=4
Tachon
A(2;1),B(4;-1).
Do I la tam ciia hinh chii nhat nen
C(7;2),
D(5;4).
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 dj: 4x + y - 9 = 0,
d2:2x-y

+ 6 = 0,
d3:x-y
+ 2 = 0. Tim toa dp cac dinh ciia hinh thoi
ABCD, biet hinh thoi ABCD c6 dien tich bang 15, cac dinh A, C
thuoc
ds, B
thuoc
di va D
thuoc
d2.
Vi
BD1 AC nen phuong
trinh
BD: y = -x + m
y = -x + m
B
= BD n di, suy ra B
^ ^ [4x + y-9 = 0
Tuong tu D = BDndj => D
•B
^9-m 4m-9^
^m-6 2m + 6^
Suy ra tga dg trung diem ciia BD la I
1
2m-l
1
2m-l
Vi
IeAC=i> —
2 2

.2 2
+
2 =
0<»m
= 3.Suyra B(2;1),D(-1;4),I
^1
5^
2'2
Cty TNHH MTV DWH
Kha„g
Viet
' C _ AT- 15 _ 5
Ta co: b^^^^ - -^ABCD - y - - ^
.AI^=^
Ma
Aedg
=> A(a;
a
+ 2) => Ar =2
1
a —
2
\
''no
nen ta c6:
1
a
2
25
oa = 3,a = -2

Vay toa do cac dinh ciia hinh thoi la:
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).
Vidu.
1.3.7.
Trong mat phMng he toa do Oxy, cho hinh thoi ABCD c6 tam
f
l\
I(2;l)
va AC = 2BD. Diem M 0;-
thuoc
duong thang AB; diem
N(0;7)
thuoc
duong thang CD . Tim toa do dinh B biet B c6 hoanh do duong.
Xffi
gidi.
Goi N' la diem doi xung ciia N qua tam I
thitaco
N'(4;-5)
va N'
thuoc
canh AB.
D
Suy ra MN' =
nen
M
phuong
trinh

AB: 4x + 3y -1 = 0 .
Vi
AC = 2BD nen AI = 2BI.
Goi H la hinh chieu ciia I len AB, ta c6:
c
B
IH
= d(I,AB) =
8 + 3-1
=
2 va
1 1
1
- +
•
IH^
lA^ IB^ 4IB2
.IB
=
IHVS
Mat
khac
B e AB =>
B(b;^—^),b
> 0 =^
IB^
= (b -2)^ +
3
Vay B(l;-1).
4b + 2

=
5ob = l
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 di, d2 va
giao
diem ciia hai duong thang do la mpt dinh ciia
Jynh
binh hanh.
Xffi
gidi
Tpa dp
giao
diem ciia dj va dj la nghi^m ciia h?: <! ^ <=>
x + y-l = 0 fx = -l
3x-y + 5 = o'^|y = 2
Phumig
pitdp
giai
Todn Hinh hoc
theo
chuyen
de-

Nguyen
Phu Khdnh,
Nguyen
Tat Thu
D
Ta gia sir
A(-l;
2) va AB = dp
AD
= d2, suy ra C(7; 4). ^
Goi
d la dtrong thang di qua I va
song song voi AB, suy ra phuang
trinh
d:x + y- 6 = 0.
Toa do giao diem ciia d va AD:
d
x+y-6=0
3x-y
+
5 = 0
<=>
1
X = —
4
23
y
=
Do do D
3 19

2' 2
, suy ra B
.M
1
^
4' 4
7^
A
B
la trung diem ciia AD
Vidi^
7.3.9.
Cho hinh binh hanh ABCD c6 B(l;5), duang cao AH:x+2y-2=0,
duong
phan
giac
trong ciia goc ACB c6 phuang
trinh
x - y -1 = 0. Tim toa
do cac
dinh
con lai ciia hinh binh hanh.
Xgigidi.
Goi
d : X - y -1 = 0.
Phuong
trinh
BC : 2x - y + 3 = 0 ,
suy ra toa do cua diem C la nghiem ciia he
2x-y+3=0

x-y-l=0
x = -4
[y
= -5^
•C(-4;-5).
H
C
Ggi
B' do'i xung voi B qua d, ta tim dugc
B'{6;0)
va B' e AC .
Suy ra phuang
trinh
AC: x - 2y - 6 = 0.
Toa dp diem A la nghiem ciia h$:
x-2y-6-0
x+2y-2=0
x = 4
A(4;-l).
Vi
AD
=
BC=>D(-1;-11).
Vidu 7.3.iO.Trong mat phang voi he toa do Oxy cho hinh vuong ABCD biet
M(2;1),N(4;-2);
P(2;0); Q(1;2) Ian
lupt
thupc canh
AB,
BC,

CD,
AD.
Hay
l^p
phuang
trinh
cac canh cua hinh vuong.
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 1NQ ".
Chung minh: Ve ME 1 CD, E € CD;
NF
1
AD,
F e AD .
Cty TNHH MTV DWH Khang Vi$t
Hai
tarn
giac
vuong MEP va NFQ c6
NF
=
ME.
Do do MP = NQ p AMEP =
ANFQ
<-> EPM = FQN
<=>

QIM = 90" o MP 1 NQ
Tra lai bai toan:
Taco:
MP = (0;-1)=>MP =
1
.
Gpi
d la duong thang di qua N va vuong
goc voi MP
Suy ra phuong
trinh
d: x - 4 = 0.
Gpi
E la giao diem cua d voi duong thang
AD,
ap dung
tinh
chat
tren ta suy ra NE = MP
Ma
E(4;m)
nen NE = MP o (m -2)^ =
1
<=>
m = 3,m = 1.
• Voi m = 3, suy ra E(4; 3) QE = (3; 1), suy ra phuang
trinh
AD:
x - 3y + 5 = 0
Phuong

trinh
AB: 3x + y - 7 = 0, BC : x - 3y -10 = 0, CD: 3x + y - 6 = 0.
• Voi m = 1, suy ra
E(4;l)
QE = (3;-l), suy ra phuang
trinh
.
AD:x
+ 3y-7 = 0 ^ ^
Phuang
trinh
AB: 3x - y - 5 = 0, BC: x + 3y + 2 = 0, CD: 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
thupc
dj,dinh
C thuoc dj va cac
dinh
B,D thupc tryc hoanh.
J^Iu&ng ddn

gidi
Vi
Aedj =>
A(t;t),
A va C doi xung nhau qua BD va B,D€Ox=>C(t;-t).
Vi
CGd2r^2t-t-l
= 0ot = l.Vay
A(l;l),C(l;-l).
flA
= IB
=
l
Trung
diem cua AC la l(l;0). Vi I la tarn cua hinh vuong nen
b-l
=1
ID
=
IA=1
b
= 0,b = 2
d
= 0,d = 2
B,D
e Ox ^ B(b;0),D(d;0) •
=^B(0;0),D(2;0)
hoac
B(2;0),D(0;0).
VayA(l;l),B(0;0),C(i;-l),D(2;0)

hoac
A(1;1),B(2;0),C(1;-1),D(0;0).
33
Phuong
fihdp
gidi
Todn Hinh hoc
thee
chuyen
de-
Nguyen
Phil Khdnh,
Nguyen
Tat Thti
Bai
1.3.2.
Trong mat phang toa do Oxy cho dirong
tron
(C):x^+y^-8x
+ 6y + 21=0 va duong thSng (d):x + y-l=0.
Xac
dinh
tga do cac
dinli
cua
hinh
vuong ABCD ngoai tiep (C) biet A e (C)
Jiuang ddn gidi
Ta
CO

I(4;-3),R
= 2 Ian lugt la tarn va ban
kinh
cua (C).
Ta
CO
led ,
hinh
vuong ABCD ngoai tie'p duong
tron
nen
lA = V2R = 2^2 . Goi A(X(,;-Xo +1) e d.
-,2
IA
= ^(X(,-4)2+(-Xo+4)2
=2V2o(xo-4)
=4=>Xo
=2,Xo=6
^ A(2;-l);C(6;-5).
Duong
thang d': x - y - 7 = 0 di qua tarn I va vuong goc voi d .
Goi
B(X(j;X() -7)ed'.
=^ IB = ^/(Xo - 4)2 + (X(, - 4)2 = 2V2 =i> XQ = 2; XQ = 6 ^ 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)

va cac hoan vi A cho C, B cho D.
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 .
Jiuang ddn glad
Ggi
C(x;y).Khid6 AB = (2;l);B(: = (x-3;y).
Tii
giac
ABCD la
hinh
vuong suy ra :
2(x-3)
+ l.y = 0
ABIBC
AB = BC
x = 4
y
= -2
hoac
x = 2
y

= 2
(x-3)%y2=5
* Voi
Ci(4;-2)=>Di(2;-3).
* Voi C2(2;2)=^D2(0;1).
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.
Jiuong ddn gidi
Vi
AClBDr^ AC:2x-y-5 = 0
Goi
I la tarn ciia
hinh
vuong => I = AC n BD =:> 1(3; 1)
=^ lA =
(-1;-2)
=^ lA = Vs =^ IB = ID = >/5
Vi
B£BDnr>B(5-2b;b)rr>iB-(2-2b;b-l)
r:> IB^ = 5 o (2b - 2)2 + (b -1)2 = 5 (b -1)2

1
c:>
b
= 0
b
= 2
34
Cty TNHH MTV DWH Khang Viet
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) * -i':
=>AB:x-3y-5
= 0, AD:3x + y-5 = 0.
Bai
1.3.5.
Trong mat phang voi he toa do Oxy, cho
hinh
chii
nhat ABCD c6
canh: AB: x - 3y + 5 = 0, duong
cheo:
BD:x-y-l = 0 va duong
cheo

AC qua
diem
M(-9; 2). Tim toa do ciic
dinh
cua
hinh
chir nhat. .i.iJ'.
J-luang ddn gidi JC
Ta
CO
toa do ciia B la nghiem cua he:
x-3y+5=0
<=>
{
x = 4
y
= 3^
x-y-l=0
BCl AB =:> BC: 3(x - 4) + (y - 3) = 0 cj. 3x + y -15 = 0 .
D
€ BD D(d;d -1) => phuong
trinh
AD: 3x + y - 4d +1 = 0 .
B(4;3).
=:> AD n AB = A :
x-3y+5=0
3x + y-4d + l = 0'
A
6d-4 2d + 7
Goi

I la tarn cua
hinh
chu nhat =^ I la
trung
diem ciia BD => I
d
+ 4 d + 2
Vi
A,
I,
M thang hang nen ta c6:
lA
=
k.MI
^
7d-28
-d + 4
d
+ 22 d-2
d
=
4=>D(4;3)
= B loai
<»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).
^ j
Bai
1.3.6.
Trong mat phang voi he toa do Oxy cho
hinh
vuong
ABCD
biet
M(2;l),
N(4;-2);
P(2;0);
Q(1;2) Ian
lupt
thupc canh
AB,
BC,
CD, AD.
Hay
lap phuong
trinh
cac canh ciia
hinh
vuong.
Jiu&ng ddn gidi
Gia su duong thSng AB qua M va c6 vec to phap tuyeh la
fi(a;
b) '
(a +b

7i
0) suy ra vec to phap tuyeh ciia BC la:nj(-b;a). ^ , ,
Phuong
trinh
AB c6 dang: ax + by - 2a - b = 0
BC
CO
dang: - bx + ay + 4b + 2a = 0
Do
ABCD
la
hinh
vuong nen
d
(P;
AB)
=
d
(Q; BC) '
Hay
-b
^£27
3b + 4a
Va2+b2
b
= -2a
b
= -a
35
Phuongphdpgidi Todn Hinh hoc

theo
chuyen de- Nguyen Phu Khdnh, NguySn Tat
Thii
•
b = -2a suy ra phirong
trinh
cac canh can tim la:
AB:x-2y
=
0; CD : x-2y-2 = 0; BC: 2x + y-6 = 0; AD: 2x + y-4 = 0.
•
b
=
-a .
Khi
do
AB:
-x
+
y
+
l = 0 ;BC: -x-y
+
2
=
0
AD:-x-y
+ 3 = 0
;CD:-x
+ y + 2 = 0.

Bai
1.3.7.
Trong mat phang voi h§ toa do Oxy cho ba diem
I(1;1),E(-2;2),
F(2;-2).
Tim tpa do cac
dinh
ciia
hinh
vuong
ABCD,
biet I la tam cua
hinh
vuong,
AB di qua E va CD di qua F.
Jiucmg
dan gidi
Duong
thang
AB
c6 phuong
trinh
dang: a(x + 2)
+
b(y - 2J = 0
«>
ax
+
by
+

2a - 2b
=
0 voi a^
+
b^
>
0.
Duong
thang CD c6 phuong
trinh
dang:
a(x
-
2)
+
b(y
+
2)
=
0 o ax
+
by - 2a
+
2b
=
0.
Vi
d(I,AB)
=
d(I,CD):

3a-b a - 3b
^b^
Suy ra phuong
trinh
AB:
x - y + 4 = 0,
CD:
x - y - 4 = 0.
Phuong
trinh
BC va DA c6 dang x + y
+
c
=
0
x
+
2
<=>
a
=
-b
d(I,BC)
=
d(I,AB)
=
2V2:
=
2^=>c = 2,c = -6.
•

BC
:
x + y + 2 = 0,
DA:
X
+ y -6 = 0
.
Suy ra
A(l;5),
B(-3;l),
C(l;-3),
D(5;l)
•
BC:x + y-6 = 0,
DA:x
+ y + 2 = 0.Suy ra
A{-3;1),
B(l;5),
C(5;l),
D(l;-3).
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 di qua
diem
M(2;l).
Tim tQa do cac
dinh
cua
hinh
chu nhat
i
Jiit&ng ddn gidi
Ta
CO
BD
n
AB
=
B(7;
3),
phuong
trinh
duong thSng
BC:
2x + y - 17
=
0
Do
Ae
AB=>A(2a

+ l;a), Ce BCC(c;17-2c), a^3, c^7,
''2a + c + l a-2c+ 17^
la
trung diem cua
AC,
BD
Suy ra I =
12 2 ,
Mal€BDo3c-a-18
= 0oa = 3c-18:
A(6c-35;3c-18)
M,
A, C thang hang
<=>
MA,MC
ciing
phuong
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
Cty
TNHH
MTV DWH Khang Viet
Bai
1.3.9-
Cho
hinh binh
hanh
ABCD
c6 di?n
tich
bang 4.
Biet
A(l;
0),
B(0;
2)
va
giao diem I
ciia
hai duong cheo nam tren duong thSng y = x. Tim toa do
dinhCvaD
,
^^^•.>P^A.^:':or.
Jiic&ng
ddn gidi
Ta

c6:
AB
= (-1;2)
=>
AB
= N/S . Phuong
trinh
cua
AB
la: 2x + y - 2 = 0.
Ie(d):y
=
x=^l(t;t).
I
la trung
diein
CLia
AC va
BD
nen ta c6: C(2t -
l;2t),
D(2t;2t
-2)
4
Mat
khac:
SJ^J^^D
=
AB.CH
- 4

(CH:
chieu cao)
=>
CH =
•
'
-
i
-'
I6t-4I
Ngoai
ra: d
(C;
AB)
=
CH
o —^ = -j=
V5
v5
<=>
t
=
-r^C
3
5_8)
3'3
,D
^8.2^
U'3.
Vay

toa do
ciia
C va D la: C
3'3,
,D
8 2
3'3
t
=
0=:>C(-!;0),D(0;-2)
hoac
C(-l;0),D(0;-2).
Bai
1.3.10.
Trong mat phang voi he toa do Oxy, cho
hinh
vuong
ABCD.
Goi
M la trung diem canh
BC,
N la diem nam tren canh CD sao cho
CN
=
2ND.
^11
n
Gia
su M
2

' 2
va
AN
:
2x - y - 3 = 0. Tim toa do diem
A.
Jiunng
ddn gidi
Gia
sir
hinh
vuong
ABCD
c6 canh la a.
Khi
do, theo de
bai,
ta c6
^2
TiOa
AN
=
\/DA^TDN^
=
2
a
a +•
AM
=
\/AB^

+BM^
=
2
a
a +-
9
3
NM
=
VcN^TcM^
=
a^
^
4a^ 5a
'4^9
6 '
Ap
dung
dinh
ly cosin cho tam giac
ANM,
ta c6:
.
AN^
+
AM^
-
NM^
V2
cos

NAM
=
=
—.
2-AN-AM
2
Do
do, phuong
trinh
duong thang AM qua M va tao voi AN mot goc
Phuang
phdpgiai
Todn Hinh
hgc
theo
chuyen
de-
Nguyen
Phi'i
Khdnh,
Nguyen
Tat Thu
Gia sir duang thang AM
c6
phap vector la ri = (a,
b)
(a^ +b'^ ^ 0). Khi do,
2a
-
b

ta
ti'nh
du-oc cos
NAM
=
VsVa^+b^'
Tu
day, do cos
NAM
=
—
nen ta
c6
A/2
|2a
-
b|
= Vs
Va^Tb^
o
3a^
-
8ab
-
3b^ = 0
<::^
a = 3b v b = -3a.
• Voi
a
= 3b : Chon

b
= 1,
a
= 3, ta
c6
AM:
X

11
1-
= 0, tiVc 3x + y-17 =
0.
Tu
day de dang tim dugc A(4,
5).
•
Voi b
=
-3a
: Chon
a
= 1, b = -3,
ta c6
AM :
1
•
(
ir
^
0

X
-3-
I
2 J
=
0, tuc
X
-
3y
-
4 = 0. Vai ke't qua nay, ta tim duoc A(l,
-1).
Vay
CO
tat ca hai diem
A
thoa man yeu cau de bai la
:
A(4,
5) va
A(l,
-1).
Bai
1.3.11.
Cho hinh thang vuong ABCD, vuong tai
A va
D. Phuang
trinh
AD
:

X
-
y
V2
= 0
.
Trung diem M cua BC
c6
toa do M(l, 0).
Biet
BC = CD = 2AB
Tim
toa do cua diem A.
Jiu&ng
dan
gidi
2 /i
Goi
H la hinh chieu cua
M len
AD ta
c6 H
3'
3
Nell
cho AB =
X
ta
c6
BC = CD = 2x de dang ta thay

MH
3x
1
X
=
•
Suy ra AD
=
BE
=
VAB^
-CE^
=
VSX
= -
Goi
A(V2t;t),suy ra HAJ^f2i ;t-~
[
3 3
2
1
Do AD =
-
nen AH =
-
hay:
3
3 ^
!-Vit
2

+
2
= i«3t2
3 9
38
Cty TNHH MTV DWIl Khnng Viet
Vay
CO
hai diein can tim la:
A
6±V6 3j2±S
g^j
1.3.12.
Trong
mat
phang Oxy
cho
bon diem M(4;5), N(6;5),
P(5;2),
Q(2;l)
•
Viet phuong
trinh
canh AB cua hinh chir nhat ABCD biet
cac
duong
thSng AB, BC, CD, DA Ian
luxit
di qua M, N,
P, Q

va dien tich hinh chir nhat
bangiR-
Jiu&ng
dan
gidi
i,,if'<')
'22
Phuong
trinh
AB
CO
dang: a(x
-
4) +
b(y-5)
0
voi
a +b
>0'
; ;
Phuang
trinh
BC : b(x
-
6)
-
a(y
-
5) =
0. , ;

4{a-3b)(a-b)
'
Di?n tich ciia hinh chu nhat:
S
= d(P, AB).d(Q, BC) =
a-fb^
Ma S = 16 nen ta
c6:
(a
-
3b)(a
-
b)
a^+b^
=
16=>
a= -1,b =
l
a^Ab.l
3
Vay phuang
trinh
AB la: x-y +
l
= 0
hoac
x-3y +11 = 0
.
Bai
1.3.13.

Trong mat phang Oxy
cho
hinh
tlioi
ABCD, phuang
trinh
hai
canh AB, AD 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
"
Toa do ciia
A
la nghiem cua he:
x+2y-2=0

_
_4
3
y
=
•
'3'3,
2x+y+l=0
Gpi
a
la canh ciia hinh
thoi,
ta suy ra:
SABCD
=
2SABD
=
2(SAMB
+
SAMD)
=
AB)
+
d(M,
AD)]
= ~
8a
Mat
khac,
S^^Q^

-
AB.AD.sinBAD = a sin
a
Ma
cos a = cos(
AB, AD)
= -
=:>
sin a = -
=> S^J^Q^
= —
5
5 5
Do do ta
c6:
,
3a^ 8a
5
S
=>a
=
sVs
B(2-2b;b), D(d;-2d-1). Tu AB = AD =
-^ ta
tim dugc
b
= -l,d =
-4
3
Vay B(4;-l),

D(-4;7)
va C
U
13^
3'
3
39
Phuong
phapgiai
Todn Hinh h(fc thco
chuyen
de-
Nguyen
Phu Khdnh,
Nguyen
Tai Thu
§ 4. CAC BAI
TO
AN
VE
Dl/CfNG
TRON VA CONIC
1.
J^fhdm
cdc bdi
todn
lien
quan
den
du&ng

tron.
Khi
giai cac bai toan ve duang tron chiing ta can luu y:
1) Vi tri
turnip
doi ^iim hai
dican^
tron
Cho hai duong tron (C,) c6 tarn I,, ban
kinh
Rj va duong tron (Cj) c6
tarn
, ban
kinh
Rj . Khi do, ta c6 cac ket qua sau:
• (C,) va (C2) khong
CO
diem chung khi va chi khi
Ijl2>R,+R2
hoac
I,l2<|R,-R2 •
• (Cj) va (C2) tiep xuc ngoai khi va chi khi Ijl2=Ri+R2-
• (Cj) va (C2) tiep xiic trong khi va chi khi I,l2 = Rj -R2 •
• (C,) va (C2) c^tnhau khi vachi khi
|R,-R2|<I,l2
<Ri+R2-
2)
Vi tri
turnip
doi ^im

ditxtn^
than;^
va
dinrnif
tron
Cho duong tron (C) c6 tam I, ban
kinh
R va duong thang A . Goi H la hinh
chieu cua I len A va dat d = IH =
d(I,
A).
Khi do:
• (C) va A khong c6 diem chung khi va chi khi d > R .
• (C) va A
CO
diing
mot diem chung khi va chi khi d.= R . Luc nay A goi la
tiep tuyen cua (C), H la tiep diem.
Chu y: Tir mot diem M nSm ngoai duong tron (C) luon ve
duoc
hai tiep
tuyeh MA, MB (A,B la cac tiep diem) den (C). Khi do MA = MB va IM la
phan
giac
ciia goc AMB.
• (C) va A
CO
diem A,B chung khi va chi khi d < R . Khi do H la trung diem
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
A(0;2),B{-2;-2),
C(4;-2).Goi
H la
chan
duong cao ke tu B; M,N Ian lugt
la trung diem cua AB, AC. Viet phuong
trinh
duong tron di qua cac diem
H,M,N.
Xgigidi.
Taco
M(-l;0),N(l;-2),AC
=
(4;-4).G9i
H(x,y),ta c6:
(BHIAC
f4(x +
2)-4(y
+ 2) = 0 fx = l
<
<=> < <=> <
HeAC l4x +
4(y-2)

= 0
y
= l
H(l;l)
40
Cty
TNHH MTV DWli Khang Vic,
Gia su phuong
trinh
duong tron: x^ + + ax + by + c = 0 .
Ba diem M,N,H thuoc duang tron nen ta c6 he phuong
trinh
:
a-c = l |'a = -l
•
a - 2b + c = -5 o
<^
b =
1
. . ;
a + b + c = -2 [c = -2
Phuong
trinh
duong tron: x^+y^-x + y- 2 = 0.
'•Yi
Vidu /.4.2.Trong mat phang voi h^ toa do Oxy, cho cho hai diem A(2;0)
va B(6; 4). Viet phuong
trinh
duong tron (C) tiep xuc voi true hoanh tai A
va khoang

each
tu tam cua (C) den diem B bang 5.
Goi
I(a;b)
va R Ian luot la tam ciia va ban
kinh
cua (C). (c ; ^ r ; t >
Vi
(C) tiep xiic voi Ox tai A nen a = 2 va R = b J ( - <*
Matkhac: IB =
5o42+(b-4f
=5^^b = l,b = 7 • • r-
2 2 ' '
Voi
b -
1
thi phuong
trinh
duong tron
(C):
(x - 2) + (y -1) = 1.
Voi
b = 7 thi phuong
trinh
duong tron
(C):
(x - if + (y - if = 49.
Vi
du
1.4.3.

Trong mat phing Oxy cho diem
M(6;6)
va hai duong thang
Aj:
4x - 3y - 24 = 0, Aj : 4x + 3y + 8 = 0. Viet phuong
trinh
duong tron (C) di
qua M va tiep xuc voi hai duong thang
Aj,
Aj.
JCffigidi.
Gpi
I(a; b) la tam va R la ban
kinh
ciia duong tron
(C).
'Ji hi ; .
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
4a-3b-24-4a
+ 3b + 8
4a-3b-24
=
-4a-3b-8

<=>
a = 2
a = 2, phuong
trinh
(C):
(x - if + (y - b)' . (^b +16)
25
2 87
Do
M e
(C)
nen (6 - 2^
+
(6 - bf =
i^^ll^
« b
=
3,
b
-
25 4
Suy ra phuong
trinh
(C):
(x - 2)^ + (y - 3)^ = 25
hoac
(C):(x-2)^.(y-f)2=^
4 16
4-
Phumtg

phdpgiai
Todn Hinh hgc
theo
chuyen de-
Nguyen
Phu Khdnh,
Nguyen
Td't Tim
•
b = - —, phuong
trinh
cua (C): (x - a)^ +
3
16
V
+ —
^
3
(4a-8)^
25
Do
M e (C) nen (6 - a)^ +
6 + —
3 j
(4a-8)^
25
phuong
trinh
v6 nghiem.
Vi

dul.4. 4. Trong mat phang vol he toa do Oxy, cho duong tron
(C):
x^ + - 2x - 2y +1 = 0 va duong thang d:x-y + 3 = 0.
Viet
phuong
trinh
duong tron (C) c6 tam M tren d, ban
kinh
bang 2 Ian ban
kinh
duong
tron
(C) va tiep xuc ngoai voi duong tron (C).
JCgigidi.
Duong
tron (C) c6 tam
1(1; 1),
ban
kinh
R = 1.
Goi
r la tam va R' la ban
kinh
ciia
duong tron (C) ta c6 R' = 2R = 2va
r
e d r(a;a + 3)
Vi
(C) va (C) tiep
xiic

ngoai voi nhau nen
11'
= R + R' = 3 ;
<=>
(a -1)^ + (a + 2)^ = 9 o a^
+
a -2 = 0 o a = l,a = -2.
•
a = l=>r(l;4)=>(C'):(x-l)^+(y-4)2 =4
•
a =
-2=>r(-2;l)=^(C'):(x
+ 2)^
+{y-lf=4.
Vidu
1.4.5.
Trong mat phang Oxy cho duong tron (C^):x^+y^-2x-2y-18=0
va duong tron
(€3):
(x + if + (y - 2)^ = 8. Chung
minh
rang hai duong tron
(Cj)
va (C2) cat nhau tai hai diem phan biet A, B.
Viet
phuong
trinh
duong
tron
(C) di qua ba diem A, B, M(0; 6).

ijpT
gidi.
Duong
tron (Cj) c6 tam
Ij(l;l),
ban
kinh
Rj =2V5 .
Duong
tron (Cj) c6 tam
l2(-l;2),
ban
kinh
R2 = 2N/2 .
•
Do 2N/5-272=RJ-RI<IJI2=%/5<RJ+R2
==275
+
2V2 nen (CJ) va (C2)
cat nhau tai hai diem phan biet A,B .
Toa dp giao diem
ciia
(Cj) va (C2) la nghiem
ciia
he:
x^
+y^ -2x-2y-18=:0
{x +
lf
+{y-2f

=8
x^ +y2
-2x-2y-18 = 0
2x.l^
= y
2 ^
x^+y2-2x-2y-18
= 0
x^
+ y^ + 2x-4y-3 = 0
y
=
2x.l^
^
2
93
5x2+24x + —= 0 (*)
4
42
Cty
TNIUl
M'lV DWH Khang
Viet
f
15
Goi x,,X2
la hai nghiem cua (*), suy ra
A|^x,;2xi
+y
Suyra

AB^ =
5(Xi -X2)^
= 5[(x, +
Xj)^
-4x,X2^
Goi
M la trung diem AB, suy ra
,B X2;2x2 +
X)
+X2 _ 12
M
2;
^
12 27^
y^
=Xi+X2+
—=
•
5 10
5
15_27
2 " 10 ; s , ,
Phuong
trinh
duong thSng
AB:
4x - 2y +15 = 0 nen
Phuong
trinh
duong trung true A cua doan

AB:
x + 2y - 3 = 0 .
Goi
I la tam cua duong tron (C), suy ra I e A => I(2a + 3; -a)
J^I'M
isT
Matkhac:
>;,>].,{
d^(I,AB).^
=
IM^o(l^.^
= (2a.3)^.(a.6)^c.a.l
Suy ra
1(5;
-1), ban
kinh
R = IM = 5^2 . ,, ij
"rm
I
Vay
phuong
trinh
cua (C): (x - 5)^ + (y +1)^ = 74.
C/iuiy Ngoai
each
giai
tren, ta c6 the sir dung chum duong tron de
giai.
Cu the:
Vi

(C) di qua cac giao diem cua (Cj) va (C2) nen phuong
trinh
cua (C)
CO
dang: m(x^ + y^ - 2x - 2y -18) + n(x^ + y^ + 2x - 4y - 3) = 0 .
Do
(C) di qua
M(0;6)
nenta
c6: 2m + 3n = 0, ta chon m = 3,n = -2
Khi
do phuong
trinh
(C): x^ + y^ - lOx + 2y - 48 = 0 .
Vi
du 1.4.6. Trong he toa do Oxy, cho duong tron (C): (x - 6)^ + (y - 2)^ = 4 .
Viet
phuong
trinh
duong tron (C) tiep xuc voi hai true tga do Ox,Oy dong
Jhoi
tiep
xiic
ngoai voi (C).
f
Xffigidi.
Duong
tron (C) c6 tam
I (6;
2), ban

kinh
R 2.
Goi
(C'):(x-af+(y-bf
=R'2 thi (C) c6 tam I'(a;b), ban
kinh
R'.
Vi
(C) tiep
xiic
voi Ox, Oy
nensuy
ra d(r,Ox) = d(r,Oy)<=>|a| = |b| = R'<=>
Hon
nua (C) tiep
xiic
voi Ox,Oy va tiep
xiic
ngoai voi (C) nen (C) nam
en phai true Oy, do do a > 0 .
43
Phuang
phiip gidi Todn Hinh
hoc
theo
chuyen
dc-
Nguyen Pltii Khduh, Nguyen
Tat
Thu

THl:
a = b
=
R=>(C'):(x-af
+(y-af
=3^
Vi
(C)
tiep xuc ngoai voi (C) nen:
^a-2
ir
= R + R'oiJ(a-6)^
+(a-2)^ =2
+
a<r>
a
=
18
Trirong
hop nay c6 2 duong tron la
:
(C;):(x-2f+(y-2f
=4
va
(q):
{x
-18)^ (y
-
isf
=

18^
TH2: a
=
-b
= R =>
(C): (x -a)^
+
(y
+
a)^ =
a^
Tuong
tu
nhu truong hop 1, ta
CO
:
ir
= R +
R'» ^(a
-
6)^
+
(a
+
2^
= 2
+
a
o
a

=
6
Vay
truong hgp nay c6
1
duong tron la (C3
j:
(x
-
6)^ +
(y
+ 6)^
= 36
.
Tom lai,
c6 3 duong tron thoa can tim
la
:
(x-2f+(y-2f
=4, (x-18)^+(y-18^=182
va
(x-6)^+(y
+
6f
=36.
Vi
du
1.4.7.
Trong mat phSng Oxy cho duong tron
(C):

(x -1)^
+
(y
-
2)^ =
9
CO
tam
I
va
diem
M(5;-3).
Chung minh rang
tu
M,
ta
c6 the ve den
(C) hai
tiep tuyen
MA, MB (A,B
la
tiep diem).
Tinh
dien
tich
ciia tu
giac
MAIB
.
JCgigidi.

Duong tron (C) c6 tam
1(1;
2),
ban
kinh
R
=
3.
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).
Ta CO
SMAIB
=
23.^,1
= lA.MA =
R.N/MI^-R^
=
3.V41-9 =12^2 (dvdt).

Vi
du
1.4.8.
Trong
mat
phang
voi he tga do
Oxy,
cho
duong tron
(C):
(x -1)^
+
(y
+
2)^
= 9
va duong thing
d
:
3x
-
4y
+
m
=
0
.
Tim m
de

tren
d
CO
duy
nhat
mpt diem
P ma tu do
c6
the ke
dupe
hai tiep tuyen PA,PB
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
Cty
TNIUI
MTV
DVVII
Kliang Viet
API = 30"
^
IP
= 2IA =
2R
=
6
.
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
hay la d(I,d)
= 6
m
=
19,m =-41.
Vi
du
1.4.9.
Cho
duong thang
A:x + y + 2 = 0va
duong tron
(C): x^
+
y^
-4x -2y
-
0. Gpi
I
la
tam
va
M thuoc duong thang
A
.

Qua
M
ke
tiep tuyen
MA,MB.
Tim M
sao
cho
di§n
tich
tu
giac
MAIB
hang 10.
(De thi DH
Khoi
A
-
2011).
JCffigidi.
Duong tron (C) c6 tam
1(2; 1),
ban
kinh
R
=
yl5=i'Al
=
S .
Matkhac S^MAI

=
2SAIBM
=5
=*iMA.IA
=
5r^MA
=
2^/5
2
Suy
ra
IM^ = lA^ + AM^ =
25.
Ma Me
A
nen
M(m;-m-2),
suy
ra
IM^ =
25 (m
-
2)
+
(m + 3)^ =
25
om^
+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
tiep tuyen
MA, MB
den
duong tron (C) voi A,B

la hai
tiep diem
sao cho
duong thSng
AB
di qua diem
E.
JCgigidi.
Duong tron (C) c6 tam 1(4;0), ban
kinh
R =
2.
Gpi M(0; m),
gia
su
T(x; y)
la tiep
<Jiem ciia tiep tuyen ve tu M toi (C).
45