Bài tập về phép biến hình – Toán 11 nâng cao
CHƯƠNG I : PHÉP DỜI HÌNH VÀ PHÉP ĐỒNG DẠNG TRONG MẶT PHẲNG
Vấn đề 1 : PHÉP DỜI HÌNH
A. KIẾN THỨC CƠ BẢN
1 Phep
�bien�h�
nh .
��N : Phep�bien�h�
nh la�
mot�quy ta�
c ���
ev �
i moi��iem
� M cua�mat�phang
� xac���
nh ����
c mot��iem
� duy nhat�
M�
cua�mat�phang
� , �iem
� M�
goi�la�
anh
� cua�M qua phe�
p bien
�h�
nh ��
o.
�K�hieu
�: f la�
mot�phep
�bien�h�
nh nao
���
o va�
M�
la�
anh
� cua�M qua phe�
p f th�ta viet�: M�
= f(M) hay
f
f(M) = M�hay f : M I��
� M�
hay M I��
� M�
. �ie�
m M goi�la��
tao anh
�.
f la�
phep
�bien
�h�
nh ��
ong nhat�� f(M) = M , M �H .
�iem
� M goi�la�
�iem
� bat���
ong , kep
�, bat�bien
�.
f1 ,f2 la�
cac�phep�bien�h�
nh th�f2 of1 la�
phep
�bien
�h�
nh .
�Neu
�H la�
mot��
h nh nao
����
o th tap
�h��
p cac��iem
� M�
= f(M), v��
i M �H, tao�thanh
� mo�
t h�
nh H�
����
c goi�la�
anh
� cua�H qua phep�bien�h�
nh f va�
ta viet�: H�
= f(H) .
2 Phep
�d���
i h nh .
�N : Phep
�d���
i h nh la�
phep
�bien
�h�
nh khong
� lam
� thay ��
oi khoang
� cach
� gi��
a hai �iem
� bat��
k , t��
c la�
v��
i
�
�
��
hai �iem
� bat��
k M,N va�
anh
� M , N cu�
a chung
� , ta luon
�co�M N = MN . ( Bao
�toan
�khoang
� cach
�).
3 T�
nh chat�: ( cua
�phep
�d���
i h nh ) .
g�L : Phep�d���
i h nh bien
�ba �iem
� tha�
ng hang
� thanh
� ba �iem
� thang
� hang
� , ba �iem
� khong
� thang
� hang
�
thanh
� ba �iem
� khong
� thang
� hang
�.
gHQ : Phep�d���
i h nh bien
�:
1. ����
ng thang
� thanh
� ����
ng thang
�.
2. Tia thanh
� tia .
3. �oan
�thang
� thanh
� �oan
�thang
� ba�
ng no�
.
4. Tam giac�thanh
� tam giac�bang
� no�
. ( Tr��
c tam
� I��
� tr�
�
c tam
� , trong
� tam
� I��
� trong
� tam
�)
5. ����
ng tron
�thanh
� ����
ng tron
�ba�
ng no�
. ( Tam
� bien
�thanh
� tam
� : I I��
� I�
, R�
=R)
6. Goc�thanh
� goc�bang
� no�
.
B . BÀI TẬP
�
x�
= 2x 1
1 Trong mpOxy cho phep
�bien
�h�
nh f : M (x;y) I��
� M�
= f(M) = �
.
�
y
� =y+3
T�
m anh
� cua
�cac
��
iem
� sau : a) A(1 ;2) b) B( 1;2) c) C(2; 4)
Giai�:
a) A�
= f(A) = (1;5)
�
b) B = f(B) = ( 7;6)
c) C�
= f(C) = (3; 1)
�
x�
= 2x y 1
2 Trong mpOxy cho phep
�bien
�h�
nh f : M (x;y) I��
� M�
= f(M) = �
.
y�
= x 2y + 3
�
T�
m anh
� cua
�cac
��
iem
� sau : a) A(2 ;1) b) B( 1;3) c) C( 2;4)
Giai�:
a) A�
= f(A) = (4;3)
b) B�
= f(B) = ( 4; 4)
c) C�
= f(C) = ( 7; 7)
3 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = (3x; y) . �ay
�co�
pha�
i la�
phep
�d�
�
i
h�
nh hay khong
�?
-1-
Bài tập về phép biến hình – Toán 11 nâng cao
Giai�: Lay
�hai �
iem
� bat��
k M(x1; y1),N(x 2 ; y 2 )
Khi ��
o f : M(x1; y1 ) I��
� M�
= f(M) = (3x1; y1) .
f : N(x 2 ; y 2 ) I��
� N�
= f(N) = (3x 2 ; y 2 )
Ta co�
: MN = (x 2 x1 )2 (y 2 y1)2 , M��
N = 9(x 2 x1)2 (y2 y1 )2
Neu
� x1 �x2 th�
M��
N �MN . Vay
�: f khong
� phai�la�
phep
�d�
��
i h nh .
(V�co�
1 so�
�
iem
� f khong
� bao
�toan
�khoang
� cach)
� .
4 Trong mpOxy cho 2 phep
�bien
�h�
nh :
a) f : M(x;y) I��
� M�
= f(M) = ( y{ ; x{ 2 )
x�
b) g : M(x;y) I��
� M�
= g(M) = ( 2x
.
{ ; y+1)
{
y�
x�
y�
Phep
�bien
�h�
nh nao
�tren
���
ay la�
phep
�d�
��
i h nh ?
HD :
a) f la�
phep
�d�
��
i h nh
b) g khong
� phai�la�
phep
�d�
��
i h nh ( v�x1 �x 2 th�
M��
N �MN )
5 Trong mpOxy cho 2 phep
�bien
�h�
nh :
a) f : M(x;y) I��
� M�
= f(M) = (y + 1 ; x)
b) g : M(x;y) I��
� M�
= g(M) = ( x ; 3y ) .
Phep
�bien
�h�
nh nao
�tren
���
ay la�
phep
�d�
��
i h nh ?
Giai�:
a) f la�
phep
�d�
��
i h nh
b) g khong
� phai�la�
phep
�d�
��
i h nh ( v�y1 �y 2 th�
M��
N �MN )
6 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = (2x ; y 1) . T�
m anh
� cu�
a�
�
�
�
ng
thang
� () : x 3y 2 = 0 qua phep
�bien
�h�
nh f .
Giai�:
Cach
� 1: Dung
� bieu
�th�
�
c toa�
��
o
� x�
�
x�
= 2x
�
x
Ta co�
f : M(x;y) I��
� M�
= f(M) = �
�� 2
y�
y 1 �
�
y y�
1
�
x�
V�M(x;y) �() � (
) 3(y�
1) 2 0 � x�
6y�
2 0 � M���
(x ;y ) �(�
) : x 6y 2 0
2
Cach
� 2 : Lay
�2 �
iem
� bat��
k M,N �() : M �N .
gM �() : M(2;0) I��
� M�
f(M) ( 4;1)
gN �( ) : N( 1; 1) I��
� N�
f(N) (2; 0)
�
gQua M�
(4;1)
x+ 4 y 1
uuuuur
(�
) �(M��
N ):�
� PTCtac�(�
):
� PTTQ (�
) : x 6y 2 0
6
1
gVTCP : M��
N (6; 1)
�
7 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = (x 3 ; y 1) .
a) CMR f la�
phep
�d�
��
i h nh .
b) T�
m anh
� cua
��
�
�
�
ng tron
�(C) : (x + 1)2 + (y 2)2 = 4 .
-2-
I��
� (C�
) : (x 2)2 + (y 3)2 = 4
Bài tập về phép biến hình – Toán 11 nâng cao
8 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = (x 3 ; y 1) .
a) CMR f la�
phep
�d�
��
i h nh .
b) T�
m anh
� cua
���
�
�
ng thang
� () : x + 2y 5 = 0 .
c) T�
m anh
� cua
��
�
�
�
ng tron
�(C) : (x + 1)2 + (y 2)2 = 2 .
x2
y2
+
=1.
3
2
Giai�: a) Lay
�hai �
iem
� bat��
k M(x1; y1),N(x2 ; y2 )
d ) T�
m anh
� cua
�elip (E) :
Khi ��
o f : M(x1;y1) I��
� M�
= f(M) = (x1 3; y1 1) .
f : N(x2 ; y2 ) I��
� N�
= f(N) = (x2 3; y 2 1)
Ta co�
: M��
N = (x 2 x1 )2 (y 2 y1)2 = MN
Vay
�: f la�
phe�
p d�
��
i h nh .
b) Cach
� 1: Dung
� bieu
�th�
�
c toa�
��
o
�
x�
= x 3 �
x x�
3
Ta co�
f : M(x;y) I��
� M�
= f(M) = �
��
y�
y 1
y y�
1
�
�
V�M(x;y) �() � (x�
3) 2(y�
1) 5 0 � x�
2y�
4 0 � M���
(x ;y ) �(�
) : x 2y 4 0
Cach
� 2 : Lay
�2 �
iem
� bat��
k M,N �() : M �N .
gM �() : M(5 ;0) I��
� M�
f(M) (2;1)
gN �() : N(3 ; 1) I��
� N�
f(N) (0;2)
�
gQua M�
(2;1)
x 2 y 1
uuuuur
(�
) �(M��
N ): �
� PTCtac�(�
):
� PTTQ (�
) : x 2y 4 0
2
1
gVTCP : M��
N (2;1)
�
Cach
� 3 : V�f la�
phep
�d�
��
i h nh nen
�f bien
���
��
ng thang
� () thanh
� ���
�
ng thang
� (�
) // ( ) .
gLay
�M �() : M(5 ;0) I��
� M�
f(M) (2;1)
�
�
gV�( ) // () � ( ) : x + 2y m = 0 (m �5) . Do : (�
) M�
(2;1) � m = 4 � (�
) : x 2y 4 0
c) Cach
� 1: Dung
� bieu
�th�
�
c toa�
��
o
�x�
= x 3 �x x�
3
Ta co�
f : M(x;y) I��
� M�
= f(M) = �
��
�
�
�y y 1
�y y 1
V�M(x;y) �(C) : (x + 1)2 + (y 2)2 = 2 � (x�
4)2 (y�
3)2 2 �
� M���
(x ;y ) �(C�
) : (x 4)2 (y 3)2 2
�
�
+ Tam
� I( 1;2) f
+ Tam
� I�
= f [ I( 1;2)] (4;3)
Cach
� 2 : (C) �
��
� (C�
) �
BK : R�
=R= 2
� BK : R = 2
�
� (C�
) : (x 4)2 (y 3)2 2
d) Dung
� bieu
�th�
�
c toa�
��
o
�
x�
= x3 �
x x�
3
Ta co�
f : M(x;y) I��
� M�
= f(M) = �
��
y�
y 1
y y�
1
�
�
x2
y2
(x�
+ 3)2
(y�
1)2
(x + 3)2
(y 1)2
���
�
V�
M(x;y) �(E) :
+
=1 �
+
= 1 � M (x ;y ) �(E ) :
+
=1
3
2
3
2
3
2
-3-
Bài tập về phép biến hình – Toán 11 nâng cao
9 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = (x 1; y 2) .
a) CMR f la�
phep
�d�
��
i h nh .
b) T�
m anh
� cua
��
�
�
�
ng thang
� () : x 2y 3 = 0.
c) T�
m anh
� cua
��
�
�
�
ng tron
�(C) : (x + 3)2 + (y 1)2 = 2 .
d) T�
m anh
� cua
�parabol (P) : y 2 = 4x .
�S : b) x 2y 2 = 0
c) (x + 2)2 + (y 1)2 = 2
d) (y + 2)2 = 4(x 1)
10 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = (x ; y) . Khang
��
�
nh nao
�sau ��
ay
sai ?
A. f la�
1 phep
�d�
��
i h nh
B. Neu
�A(0 ; a) th�
f(A) = A
C. M va�
f(M) ���
oi x �
ng nhau qua truc�hoanh
�
D. f [M(2;3)] ��
�ng
�
� thang
� 2x + y + 1 = 0
�S : Chon
�C . V�
M va�
f(M) ���
oi x �
ng nhau qua truc�tung � C sai .
12 Trong mpOxy cho 2 phep
�bien
�h�
nh :
f1 : M(x;y) I��
� M�
= f1(M) = (x + 2 ; y 4) ; f2 : M(x;y) I��
� M�
= f2 (M) = ( x ; y) .
T�
m toa�
��
o anh
� cua
�A(4; 1) qua f1 roi�f2 , ngh�
a la�
t�
m f2 [f1(A)] .
f
f
1 � A�
2 � A�
�
�S : A(4; 1) I��
(6; 5) I��
(6 ; 5 ) .
x
11 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = ( ; 3y) . Khang
� ��
nh na�
o sau ��
ay sai ?
2
A. f (O) = O (O la�
�iem
� bat�ien)
b �
B. Anh
� cua
�A �Ox th�anh
� A�
= f(A) �Ox .
C. Anh
� cua
�B �Oy th��
anh B�
= f(B) �Oy .
D. M�
= f [ M(2 ; 3)] = (1; 9)
�S : Chon
�D . V�M�
= f [ M(2 ; 3)] = (1; 9)
Vấn đề 2 : PHÉP TỊNH TIẾN
A. KIẾN THỨC CƠ BẢN
uuuuu
r r
r
1 �N : Phep
��
t nh tien
�theo vect�u la�
mot�phep
�d�
��
i h nh bien
��
iem
� M tha�
nh �
iem
� M�
sao cho MM�
u.
uuuuu
r r
K�hieu
�: T hay Tur .Khi ��
o : Tur (M) M�
� MM �
u
gPhep
��
t nh tien
�hoan
�toan
��
�
�
�
c xac��
�
nh khi biet�vect��
t nh tien
�cua
�no �
.
r
r
gNeáu To (M) M , M thì To laø pheùp ñoàng nhaát .
r
2 Bieåu thöùc toïa ñoä : Cho u = (a;b) vaø pheùp tònh tieán Tur
�x�
=x+a
M(x;y) I��
� M�
=Tur (M) (x��
; y ) th��
�
�y = y + b
3 T�
nh chat�:
g�L : Phep
��
t nh tien
�bao
�toan
�khoang
� c ach
� gi�
�
a hai �
iem
� bat��
k .
gHQ :
1. Bao
�toan
��
t nh thang
� hang
� va��
th �
t�
�
cua
�cac��
iem
� t��
ng ��
ng .
2. Bien
�mot�tia thanh
� tia .
3. Bao
�toan
��
t nh thang
� hang
� va��
th �
t�
�
cua
�cac��
iem
� t��
ng �
�
ng .
5. Bien
�mot��
oan
�thang
� thanh
��
oan
�thang
� bang
� no�
.
6. Bien
�mot���
�
�
ng thang
� thanh
� mot��
��
�
ng thang
� song song hoa�
c trung
� v�
�
i�
�
��
ng thang
� ��
a cho .
7. Bien
�tam giac
�thanh
� tam giac�bang
� no�
. (Tr�
�
c tam
� I��
� tr�
�
c tam
� , trong
� tam
� I��
� trong
� tam
�)
8. ��
�
�
ng tron
�thanh
��
�
�
�
ng tron
�ba�
ng no�
.
(Tam
� bien
�thanh
� tam
� : I I��
� I�
, R�
=R)
PHƯƠNG PHÁP TÌM ẢNH CỦA MỘT ĐIỂM
-4-
Bài tập về phép biến hình – Toán 11 nâng cao
�x�
=x+a
M(x;y) I��
� M�
=Tur (M) (x��
; y ) th��
=y+b
�y�
PHƯƠNG PHÁP TÌM ẢNH CỦA MỘT HÌNH (H) .
Cach
� 1 : Dung
� t�
nh chat�(cung
� ph�
�g
n cua
��
thang
� , ban
�k�
nh �
��
�
ng tron
�: khong
� ��
oi )
1. Lay
�M ��
(H) I
M� (H�
)
2. g(H) ���
�
�
ng thang
� ��
� (H�
) ��
�
�
�
ng tha�
ng cung
� ph�
�
ng
�
�
Tam
�I
Tam
� I�
g(H) �(C) �
I��
� (H�
) �(C�
)�
(can
�t�
m I�
).
+ bk : R
+ bk : R�
=R
�
�
Cach
� 2 : Dung
� bieu
�th�
�
c toa
���
o.
T�
m x theo x�
, t�
m y theo y�
roi�hay
t
vao
�bieu
�th�
�
c toa
���
o.
Cach
� 3 : Lay
�hai �
iem
� phan
�biet�: M, N ��
(H) I
M�
, N� (H�
)
B, BÀI TẬP
r
1 Trong mpOxy . T�
m anh
� cua
�M�
cua
��
ie�
m M(3; 2) qua phep
��
t nh tien
�theo vect�u = (2;1) .
Giai�
uuuuu
r r
�
�
x�
3 2
x�
5
Theo �
�
nh ngh�
a ta co�
: M�
= Tur (M) � MM�
u � (x �
3; y�
2) (2;1) � �
��
y�
2 1
y�
1
�
�
� M�
(5; 1)
r
2 T�
m anh
� cac��
iem
� ch�
ra qua phep
�t�
nh tien
�theo vect�u :
r
a) A( 1;1) , u = (3;1)
� A�
(2;3)
r
b) B(2;1) , u = ( 3;2)
� B�
( 1;3)
r
c) C(3; 2) , u = ( 1;3)
� C�
(2;1)
r
3 Trong mpOxy . T�
m anh
� A��
,B lan
��
l ��
t cua
��iem
� A(2;3), B(1;1) qua phep
��
t nh tien
�theo vect�u = (3;1) .
uuur uuuur
T�
nh ��
o dai�AB , A��
B .
Giai�
uuur
uuuur
Ta co�
: A�
= Tur (A) (5; 4) , B�
= Tur (B) (4;2) , AB = |AB | 5 , A��
B = |A ��
B | 5 .
r r
r
4 Cho 2 vect�u1; u2 . G�
a s��
M1 Tur (M),M 2 Tur (M1). T�
m v ��
e M2 Tvr (M) .
1
2
Giai�
uuuuur r
uuuuuuur r
r
r
Theo ��
e : M1 Tu (M) � MM1 u1 , M2 Tu (M1) � M1M2 u2 .
1 uuuuuu
r r r uuuuuur 2uuuuur uuuuuuur r r
r r r
r
Neu
�: M2 Tv (M) � MM2 v � v MM2 MM1 M1M2 u1+ u2 .Vay
�: v u1 + u2
5 ��
�
�
ng thang
� cat�Ox tai�A( 1;0) , c at�Oy tai�B(0;2) . Hay
�viet�ph�
�
ng tr�
nh �
�
�
�
ng thang
� �
la�
anh
�
r
cua
� qua phep
��
t nh tien
�theo vect�u = (2; 1) .
-5-
Bài tập về phép biến hình – Toán 11 nâng cao
Giai� V�
: A�
Tur (A) (1; 1) , B�
Tur (B) (2;1) .
�
gqua A�
(1;uuuu
1)
ur
Mat�khac�: �
Tur () � �
�
i qua A ��
,B . Do ��
o : �
�
��
g
VTCP
:
A
B
= (1;2)
�
�
x 1 t
� ptts �
:�
y 1 2t
�
6 ��
�
�
ng thang
� cat�Ox tai�A(1;0) , c at�Oy tai�B(0;3) . Hay
�viet�ph�
�
ng tr�
nh �
�
�
�
ng thang
� �
la�
anh
�
r
cua
� qua phep
��
t nh tien
�theo vect�u = ( 1; 2) .
Giai�
r (A) (0; 2) , B�
r (B) (1;1) .
V�
: A�
Tu
Tu
�
gqua A�
(0; 2)
�
x t
r ( ) � �
uuuuu
r
Mat�khac
�: �
Tu
�
i qua A ��
,B . Do ��
o : �
� ptts �
:�
�
y 2 3t
gVTCP : A�
B�
= ( 1;3)
�
�
r
7 T�
�
ng t�
�
: a) : x 2y 4 = 0 , u = (0 ; 3)
� �
: x 2y 2 0
r
b) : 3x y 3 = 0 , u = ( 1 ; 2)
� �
: 3x y 2 0
r
8 T�
m anh
� cua
��
�
�
�
ng tron
�(C) : (x + 1)2 (y 2)2 4 qua phep
��
t nh tien
�theo vect�u = (1; 3) .
Giai�
�
�
x�
=x+1
x = x�
1
r la�
Bieu
�th�
�
c toa�
��
o cua
�phep
��
t nh tien
�Tu
: �
��
y�
= y 3
y = y�
+3
�
�
2 (y�
V�: M(x;y) �(C) : (x + 1)2 (y 2)2 4 � x�
1)2 4 � M���
(x ;y ) �(C�
) : x 2 (y 1)2 4
Vay
�: Anh
� cua
�(C) la�
( C�
) : x2 (y 1)2 4
9 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = (x 1; y 2) .
a) CMR f la�
phep
�d�
��
i h nh .
b) T�
m anh
� cua
��
�
�
�
ng thang
� () : x 2y 3 = 0.
c) T�
m anh
� cua
��
�
�
�
ng tron
�(C) : (x + 3)2 + (y 1)2 = 2 .
d) T�
m anh
� cua
�parabol (P) : y 2 = 4x .
�S : b) x 2y 2 = 0
c) (x + 2)2 + (y 1)2 = 2
d) (y + 2)2 = 4(x 1)
10 Trong mpOxy cho phep
�bien
�h�
nh f : M(x;y) I��
� M�
= f(M) = (x ; y) . Khang
��
�
nh nao
�sau ��
ay
sai ?
A. f la�
1 phep
�d�
��
i h nh
B. Neu
�A(0 ; a) th�f(A) = A
C. M va�
f(M) ���
oi x �
ng nhau qua truc�hoanh
�
D. f [ M(2;3)] ��
�
�
�
ng thang
� 2x + y + 1 = 0
�S : Chon
� C . V�M va�
f(M) ���
oi x �
ng nhau qua truc�tung � C sai .
r
9 T�
m anh
� cua
��
�
�
�
ng tron
�(C) : (x 3)2 ( y 2)2 1 qua phep
��
t nh tien
�theo vect�u = ( 2;4) .
�
�
x�
= x2
x = x�
+2
Giai�: Bieu
�th�
�
c toa�
��
o cua
�phe�
p t�
nh tien
�Tur la�
: �
��
�
�
y
=
y
4
y
=
y
4
�
�
V�: M(x;y) �(C) : (x 3)2 (y 2)2 1 � (x�
1)2 (y�
2)2 1 � M���
(x ;y ) �(C�
) : (x�
1)2 (y�
2)2 1
Vay
�: Anh
� cua
�(C) la�
(C�
) : (x 1)2 (y 2)2 1
-6-
Bài tập về phép biến hình – Toán 11 nâng cao
r
BT T��
ng t�
�
: a) (C) : (x 2)2 (y 3)2 1, u = (3;1)
r
b) (C) : x2 y2 2x 4y 4 0, u = ( 2;3)
� (C�
) : (x 1)2 (y 2)2 1
(C�
) : x2 y2 2x 2y 7 0
10 Trong he�
truc�toa�
��
o Oxy , xac��
�
nh toa�
��
o cac��
�
nh C va�
D cua
�h�
nh b�
nh hanh
� ABCD biet��
�
nh
A( 2;0), �
�
nh B( 1;0) va�
giao �
iem
� cac��
�
�
�
ng cheo
�la�
I(1;2) .
Giai�
uur
uur
uur
gGoi�C(x;y) .Ta co�
: IC (x 1; y 2),AI (3;2) ,BI (2; 1)
gV�I la�
trung �
iem
� cua
�AC nen
�:
uur uur
�x 1 3
�
x4
C = Tuur (I) � IC AI � �
��
� C(4; 4)
AI
y4
�y 2 2
�
gV�I la�
trung �
iem
� cua
�AC nen
�:
uur uur
x 1 2
x 3
�
�
D = Tuur (I) � ID BI � � D
� �D
� D(3; 4)
BI
yD 2 2
yD 4
�
�
Bai�tap
��
t �
ng t�
�
: A( 1;0),B(0;4),I(1;1)
� C(3;2),D(2; 2) .
11 Cho 2 �
�
��
ng thang
� song song nhau d va�
d�
. Hay
�ch�ra mot�phep
��
t nh tien
�
bien
�d thanh
� d�
. Hoi�co�
bao nhie�
u phep
��
t nh tien
�nh�the�
?
Giai�: Chon
�2 �
iem
� co�
�
�
nh A �d , A�
�d �
uuuuu
r uuur
Lay
��
iem
� tuy�
y�
M �d . G�
a s�
�
: M�
= Tuuur (M) � MM�
AB
AB
uuuu
r uuuur
� MA M�
B � M�
B / /MA � M�
�d�
� d�
= Tuuur (d)
AB
Nhan
�xet�: Co�
vo�
so�
phep
��
t nh tien
�bien
�d thanh
� d�
.
12 Cho 2 �
�
�
�
ng tron
�(I,R) va�
(I�
,R�
) .Hay
�ch�ra mot�phep
��
t nh
tien
� (I�
,R�
).
uuu
uu
r�bien
uu
r �(I,R) thanh
u
u
r
Giai�: Lay
��
iem
� M tuy�
y�
tren
�(I,R) . G�
a s�
�
: M�
= T (M) � MM�
II�
II�
uuu
r uuuur
r [(I,R) ]
� IM I��
M � I��
M IM R � M�
�(I�
,R�
) � (I�
,R�
) = Tuu
II�
13 Cho h�
nh b�
nh hanh
� ABCD , hai �
�
nh A,B co�
�
�
nh , tam
� I thay ��
oi di ��
ong
tren
��
�
�
�
ng tron
�(C) .T�
m quy�ch
t� trung �
iem
� M cua
�canh
� BC.
Giai�
uuu
r uur
Goi�J la�
trung �
iem
� canh
� AB . Khi �
o�
de�
thay
�J co�
�
�
nh va�
IM JB .
Vay
�M la�
anh
� cua
�I qua phep
��
t nh tien
�Tuur . Suy ra : Quy�
t�
ch cua
�M la�
JB
uur
anh
� cua
��
�
�
�
ng tron
�(C) trong phep
�t�
nh tien
�theo vect�JB
-7-
Bài tập về phép biến hình – Toán 11 nâng cao
r
14 Trong he�
truc�toa�
��
o Oxy , cho parabol (P) : y = ax2 . Goi�T la�
phep
��
t nh tien
�theo vect�u = (m,n)
va�
(P�
) la�
anh
� cua
�(P) qua phe�
p t�
nh tien
���
o . Hay
�viet�ph�
�
ng tr�
nh cua
�(P�
).
Giai�:
uuuuu
r r
uuuuu
r
Tur
gM(x;y) I���
� M���
(x ;y ) , ta co�
: MM �
= u , v��
i MM�
= (x�
x ; y�
y)
uuuuu
r r
�
�
x�
x = m
x = x�
m
V�MM�
=u��
��
y�
y = n
y = y�
n
�
�
Ma�
: M(x; y) �(P) : y ax 2 � y�
n = a(x �
m)2 � y �
= a(x�
m)2 n � M���
(x ;y ) �(P �
) : y = a(x m)2 n
Vay
�: Anh
� cua
�(P) qua phep
��
t nh tie �
n Tur la�
(P�
) : y = a(x m)2 n � y = ax 2 2amx am 2 n .
r r
15 Cho �
t : 6x + 2y 1= 0 . T�
m vect�u �0 ��
e = Tur ( ) .
r
r
r
r
Giai�: VTCP cua
� la�
a = (2; 6) . �e�
: = Tur ( ) � u cung
� ph��
ng a . Khi ��
o : a = (2; 6) 2(1; 3)
r
� chon
�u = (1; 3) .
r
r
16 Trong he�
truc�toa�
��
o Oxy , cho 2 �
iem
� A( 5;2) , C( 1;0) . Biet�: B = Tur (A) , C = Tvr (B) . T�
m u va�
v
��
e co�
the��
th �
c hien
�phep
�bie �
n ��
oi A thanh
�C?
Giai�
Tur
Tvr
A( 5;2) I���
� B I��
�
� C(1; 0) .
uuur r uuur r uuur uuur uuur r r
Ta coù : AB u, BC v � AC AB BC u v (4; 2)
Tur + vr
-8-
Bài tập về phép biến hình – Toán 11 nâng cao
r
r
17 Trong he�
truc�toa�
��
o Oxy , cho 3 �
iem
� K(1;2) , M(3; 1),N(2; 3) va�
2 vect�u = (2;3) ,v = ( 1;2) .
T�
m anh
� cua
�K,M,N qua phep
��
t nh tien
�Tur roi�Tvr .
uuur r uuur r uuur uuur uuur r r
Tur
Tvr
HD : G�
a s�
�
: A(x;y) I���
� B I���
� C(x��
; y ) . Ta co�
: AB u, BC v � AC AB BC u v (1;5)
uuuur
�
�
x�
1 1
x�
2
Do ��
o : K�
=Tur vr (K) � KK�
(1;5) � �
��
� K�
(2;7) .
y�
2 5 �
y�
7
�
T�
�
ng t�
�
: M�
(4;4) , N�
(3;2) .
18 Trong he�
tru�
c toa�
��
o Oxy , cho ABC : A(3;0) , B( 2;4) , C( 4;5) . G la�
trong
� tam
� ABC va�
phep
�
r r
r
t�
nh tien
�theo vect�u �0 bien
�A than
�h G . T�
m G�
= Tu (G) .
Giai�
Tur
Tur
A(3;0) I���
� G(1;3) I���
� G���
(x ; y )
uuur
u
u
u
u
r
r
r
�
�
x�
1 4
x�
5
V�AG (4;3) u . Theo ��
e : GG�
u��
��
� G�
(5;6).
�
�
y 3 3
y 6
�
�
19 Trong mat�phang
� Oxy , cho 2 �
�
�
�
ng tron
�(C) : (x 1)2 (y 3)2 2,(C�
) : x2 y 2 10x 4y 25 0.
r
Co�
hay khong
� phe��
p t nh tien
�vect�u bien
�(C) thanh
� (C�
).
HD : (C) co��
tam I(1; 3), ban
�k�
nh R = 2 ; (C�
) co��
tam I�
(5; 2), ban
�k�
nh R�
=2.
r
Ta thay
�: R = R�
= 2 nen
�co�
phep
�t�
nh tien
�theo vect�u = (4;1) bien
�(C) thanh
� (C�
).
20 Trong he�
truc�toa�
��
o Oxy , cho h�
nh b�
nh hanh
� OABC v�
�
i A( 2;1) va�
B � :2x y 5 = 0 . T�
m tap
�
h�
�
p�
�
nh C ?
Giai�
uuur uuur
r
gV�OABC la�
h�
nh b�
nh hanh
� nen
�: BC AO (2; 1) � C Tur (B) v�
�
i u = (2; 1)
uuur r
Tur
�
�
x�
x 2
x x�
2
gB(x;y) I���
� C(x��
; y ) . Do : BC u � �
��
y�
y 1 �
y y�
1
�
gB(x;y) � � 2x y 5 = 0 � 2x�
y�
10 = 0 � C(x��
; y ) � �
: 2x y 10 = 0
21 Cho ABC . Goi�A1,B1,C1 lan
��
l �
�
t la�
trung �
iem
� cac�canh
� BC,CA,AB. Goi�O1,O2 ,O3 va�
I1,I2 ,I3
t�
�
ng �
�
ng la�
cac�tam
��
��
�
ng tro�
n ngoai�tiep
�va�
cac�tam
��
�
�
�
ng tro�
n noi�tiep
�cua
�ba tam giac�AB1C1,
BC1A1, va�
CA1B1 . Ch�
�
ng minh rang
� : O1O2O3 I1I 2I3 .
HD :
wXet�phep
��
t nh tien
�: T1 uuur bien
�A I��
� C,C1 I��
� B, B1 I��
� A1 .
AB
2
T1 uuur
T1 uuur
T1 uuur
AB
AB
AB
2
2
2
� AB1C1 I����
� C1BA1;O1 I����
� O2 ;I1 I����
� I2 .
uuuuuur uuuur
� O1O2 I1I2 � O1O2 I1I2 .
wLy�
luan
��
t �ng t�
�
: Xet�cac
�phep
�t�
nh tien
�T1 uuur ,T1 uuur suy ra :
BC
CA
2
2
uuuuuur uuuur
uuuuuu
r uuuu
r
O2O3 I2 I3 va�
O3O1 I3I1 � O2O3 I2 I3 ,O3O1 I3I1 � O1O2O3 I1I2 I3 (c.c.c).
� 60o,B
� 150ova�
� 90o.
22 Trong t�
�
giac�ABCD co�
AB = 6 3cm ,CD 12cm , A
D
T�
nh ��
o dai�cac�canh
� BC va�
DA .
HD :
uuuu
r uuur
Tuuur
� 30o(v�B
� 150o)
BC � M � AM BC.Ta co�
wXet�: A I���
: ABCM la�
h�
nh b �
nh hanh
� va�
BCM
-9-
Bài tập về phép biến hình – Toán 11 nâng cao
� 360o (90o 60o 150o) 60o � MCD
� 30o.
Lai�co�
: BCD
��
nh ly�
ham
� cos trong MCD :
3
MD2 MC2 DC2 2MC.DC.cos30o (6 3)2 (12)2 2.6 3.12.
36
2
� MD = 6cm .
1
Ta co�
: MD = CD va�
MC = MD 3 � MDC la�m
ta giac���
eu
2
� 90o va�
�
� MCD la�
n�
�
a tam giac���
eu � DMC
MDA
30o.
�
�
� 30o � AMD la�
Vay
�: MDA
MAD
MAB
tam giac�ca�
n tai�M .
6 3
Döïng MK AD � K laø trung ñieåm cuûa AD � KD=MDcos30o
cm � AD 6 3cm
2
Toùm laïi : BC = AM = MD = 6cm , AD = AB = 6 3cm
Vấn đề 3 : PHÉP ĐỐI XỨNG TRỤC
A , KIẾN THỨC CƠ BẢN
1 �N1: �iem
� M�
goi�la�
���
oi x �
ng v�
�iem
i � � M qua �
�
�
�
ng thang
� a neu
�a la�
��
�
�
ng trung tr�
�
c cua
��
oan
�
MM�
.
Phep
����
oi x �
ng qua �
�
�
�
ng tha�
ng con
�goi�la�
phep
����
oi x �
ng truc�. ��
�
�
ng thang
� a goi�la�
tru�
c ���
oi x �
ng.
�N2 : Phep
����
oi x �
ng qua �
��
�
ng tha�
ng a la�
phep
�bien
�h�
nh bien
�moi��
iem
� M thanh
��
iem
� M�
���
oi x �
ng
v�
�
i M qua �
�
�
�
ng tha�
ng a .
uuuuuur
uuuuuu
r
K�hieu
�: �a (M) M�
� M o M�
M oM , v��
i Mo la�
h�
nh chieu
�cua
�M tren
��
�
��
ng thang
�a.
Khi đó :
gNeu
�M �a th��a (M) M : xem M la�
���
oi x �
ng v�
�
i ch�
nh no�
qua a . ( M con
�goi�la�
�
iem
� bat���
ong )
gM �a th��a (M) M�
� a la�
����
ng trung tr��
c cu�
a MM�
gÑa (M) M�
thì Ña (M�
)M
gÑa (H) H�
thì Ña (H�
) H , H�
laø aûnh cuûa hình H .
g�N : d la�
truc����
oi x �
ng cua
�h�
nh H � �d (H) H .
gPhep
����
oi x �
ng truc
�hoan
�toan
�xac��
�
nh khi biet�truc����
oi x �
ng cua
�no�
.
Chu�
y�
: Mot��
h nh co�
the�
khong
� co�
truc����
oi x �
ng ,co�
the�
co�
mot�hay nhieu
�truc����
oi x �
ng .
2 Bieu
�th��
c toa
���
o : M(x;y) I��
� M�
�d (M) (x��
;y )
�x�
�x�
=x
=x
�d �Ox : �
�d �Oy : �
�
�
y
=
y
y
�
� =y
3 �L : Phep
����
oi x �
ng truc�la�
mot�hep
p �d�
��
i h nh .
gHQ :
1.Phep
����
oi x �
ng truc�bien
�ba �
iem
�thang
� hang
� thanh
� ba �
iem
� thang
� han
�g va�
bao
�toan
�th�
�
t�
�
cua
�cac�
�
iem
� t�
�ng �
�
ng .
2. ��
�
�
ng thang
� thanh
��
�
�
�
ng thang
�.
3. Tia thanh
� tia .
4. �oan
�thang
� thanh
��
oan
�thang
� ba�
ng no�
.
5. Tam giac�thanh
� tam giac�bang
� no�
. (Tr�
�
c tam
�I��
� tr�
�
c tam
� , tron
�g tam
�I��
� trong
� tam
�)
6. ��
�
�
ng tron
�thanh
��
�
�
�
ng tron
�ba�
ng no�
. (Tam
� bien
�thanh
� tam
� : I I��
� I�
, R�
=R)
7. Goc�thanh
� goc�bang
� no�
.
- 10 -
Bài tập về phép biến hình – Toán 11 nâng cao
�PP : Tìm aûnh M�
= Ña (M)
1. (d) M , d a
2. H = d �a
3. H laø trung ñieåm cuûa MM�
� M�
?
�PP : T�
m anh
� cua
��
�
�
�
ng thang
� : �
= �a ( )
wTH1: () // (a)
1. Lay
�A,B �() : A �B
2. T�
m anh
� A�
= �a (A)
3. �
A��
, // (a) � �
w TH2 : // a
1. T�
m K = �a
2. Lay
�P � : P �K .T�
m Q = �a (P)
3. �
�(KQ)
m M �() : (MA + MB)min .
�PP : T�
T�
m M �() : (MA+ MB)min
wLoai�1 : A, B nam
� cung
� ph����
a oi v �
i ( ) :
1) goi�A�
la�
���
oi x �
ng cua
�A qua ()
2) M �(), th�MA + MB MA�
+ MB �A�
B
Do ��
o: (MA+MB)min= A�
B � M = (A�
B) �()
wLoai�2 : A, B nam
� khac�ph����
a oi v �
i ( ) :
M �( ), th�MA + MB �AB
Ta co:�(MA+MB)min = AB � M = (AB) �()
B . BÀI TẬP
1 Trong mpOxy . T�
m anh
� cua
�M(2;1) ��
o i x�
�
ng qua Ox , roi����
oi x �
ng qua Oy .
�
�
Oy
Ox � M�
�
HD : M(2;1) I���
(2; 1) I���
� M�
(2; 1)
2 Trong mpOxy . T�
m anh
� cua
�M(a;b) ��
o i x�
�
ng qua Oy , roi����
oi x �
ng qua Ox .
�
�
Oy
Ox � M�
�
HD : M(a;b) I���
� M�
( a;b) I���
(a; b)
�
�
a�
�
3 Cho 2 �
�
�
�
ng thang
� (a) : x 2 = 0 , (b) : y + 1 = 0 va�
�
iem
� M( 1;2) . T�
m : M I��
� M�
I��b�
� M�
.
�
�
a�
�
HD : M( 1;2) I��
� M�
(5;2) I��b�
� M�
(5; 4) [ ve�
h�
nh ] .
4 Cho 2 �
�
�
�
ng thang
� (a) : x m = 0 (m > 0) , (b) : y + n = 0 (n > 0).
�
�
a�
b � M�
�
��
T�
m M�
: M(x;y) ��
� M���
(x ; y ) ���
(x ��
;y�
).
�
�a
�b
�
�
x�
2m x
x�
2m x
�
HD : M(x;y) I����
� M�
I������
� M�
��
��
t�
(m;y)
t
�
(
2m
x;
n)
y y
y�
2n y
�
�
5 Cho �
iem
� M( 1;2) va�
�
�
�
�
ng thang
� (a) : x + 2y + 2 = 0 .
HD : (d) : 2x Ǯy+�4
= 0 , H = d a H( 2;0) , H la�
trung �
iem
� cua
�MM� M�
( 3; 2)
6 Cho �
iem
� M( 4;1) va�
�
�
�
�
ng thang
� (a) : x + y = 0 .
� M�
= �a (M) (1; 4)
7 Cho 2 �
�
�
�
ng thang
� () : 4x y + 9 = 0 , (a) : x y + 3 = 0 . T�
m anh
� �
= �a () .
HD :
4 1
gV� � ��
cat
�aǮ K
a K( 2;1)
1 1
gM( 1;5) � � d M, a � d : x y 4 0 � H(1/ 2; 7 / 2) : t�
iem
� c ua
�MM�
� M�
�a (M) (2;2)
g�
�KM�
: x 4y + 6 = 0
- 11 -
Bài tập về phép biến hình – Toán 11 nâng cao
8 T�
m b = �a (Ox) v�
�
i�
�
�
�
ng thang
� (a) : x + 3y + 3 = 0 .
HD : ga �Ox = K( 3;0) .
3 9
gM �O(0;0) �Ox : M�
= �a (M) = ( ; ) .
5 5
�
gb �KM : 3x + 4y 9 = 0 .
9 T�
m b = �a (Ox) v�
�
i�
�
�
�
ng thang
� (a) : x + 3y 3 = 0 .
HD : ga �Ox = K(3;0) .
gP �O(0;0) �Ox .
�
+ Qua O(0;0)
g �
� : 3x y 0
+ a
�
3 9
3 9
gE = a � � E( ; ) la�
trung �
iem
�OQ � Q( ; ) .
10 10
5 5
gb �KQ : 3x + 4y 9 = 0 .
10 T �
m b = �Ox (a) v�
�
i�
�
�
�
ng thang
� (a) : x + 3y 3 = 0 .
Giai�:
Cach
� 1: Dung
� bieu
�th�
�
c toa�
��
o (rat�hay)
Cach
� 2 : gK= a Ǯ Ox K(3;0)
gP(0;1) �a � Q = �Ox (P) = (0; 1)
gb �KQ : x 3y 3 = 0 .
11 Cho 2 �
�
�
�
ng thang
� () : x 2y + 2 = 0 , (a) : x 2y 3 = 0 . T�
m anh
� �
= �a ( ) .
PP : / /a
Cach
� 1 : T�
m A,B � � A��
,B ��
� �
�A�
B�
Cach
� 2 : T�
m A � � A�
��
� �
/ / , �
A�
Giai�
: gA(0;1) � � A�
�a (A) (2; 3)
g�
A��
, / / � �
: x 2y 8 0
12 Cho �
�
�
�
ng tron
�(C) : (x+3)2 (y 2)2 1 , �
�
�
�
ng thang
� (a) : 3x y + 1= 0 . T�
m (C�
) = �a [(C)]
HD : (C�
) : (x 3)2 y2 1 .
13 Trong mpOxy cho ABC : A( 1;6),B(0;1) va�
C(1;6) . Khang
��
�
nh nao
�sau ��
ay sai ?
A. ABC can
��
�
B
B. ABC co�
1 truc���
oi x�
�
ng
C. ABC �Ox (ABC)
D. Trong
� tam
� : G = �Oy (G)
HD : Chon
�D
14 Trong mpOxy cho �
iem
� M( 3;2), ����
ng thang
� () : x + 3y 8 = 0, ����
ng tron
�(C) : (x+3)2 (y 2)2 4.
T�
m anh
� cua
�M, () va�
(C) qua phe�
p ���
oi x �
ng truc�(a) : x 2y + 2 = 0 .
Giai�: Goi�M�
, ( �
) va�
(C�
) la�
anh
� cua
�M, () va�
(C) qua phep
����
oi x �
ng truc�a .
�
g Qua M( 3;2)
a) T�
m anh
� M�
: Goi�����
ng thang
� (d) : �
ga
�
+ (d) (a) � (d) : 2x y + m = 0 . V�(d) M( 3 ;2) � m = 4 � (d) : 2x y 4 = 0
- 12 -
Bài tập về phép biến hình – Toán 11 nâng cao
�
1
x H (x M x M�
)
�
2
�
+ H = (d) �(a) � H( 2;0) � H la�
trung �
iem
� cu �
a M,M � H �
1
�
y H (y M y M�
)
�
2
�
1
2 (3 x M�
)
�
�
x
1
2
��
� � M�
� M�
( 1; 2)
1
y
2
�
�
M
�
0 (2 y M�
)
� 2
b) T�
m anh
� ( �
):
1
3
gV� � � ( ) cat�
(a) � K= ( ) �(a)
1 2
�
x + 3y 8 = 0
� Toa�
��
o cua
�K la�
nghiem
� cua
�he �
: �
� K(2; 2)
x 2y + 2 = 0
�
gLay
�P �K � Q = �a [P( 1;3)] = (1; 1) . ( Lam
� t�
�
ng t�
�
nh�cau
� a) )
�
g Qua P( 1;3)
Goi��
�
�
�
ng thang
� (b) : �
g a
�
+ (b) (a) � (b) : 2x y + m = 0 . V�
(b) P( 1;3) � m = 1 � (b) : 2x y 1 = 0
+ E = (b) �(a) � E(0;1) � E la�
trung �
ie�
m cua
�P,Q �
�
� 1
1
x (x xQ )
0 (1 xQ )
�
xQ 1
�
�
�E 2 P
�
�
� E�
�� 2
��
� Q(1; 1)
1
1
y Q 1
�
�
�
y (y yQ )
1 (3 y Q )
�E 2 P
� 2
�
gQua K(2;2)
x2 y2
uuur
+ ( �
) �(KQ) : �
� ( �
):
� 3x y 4 0
1
3
gVTCP : KQ (1; 3) (1;3)
�
c) + T�
m anh
� cua
�tam
� I( 3;2) nh�cau
� a) .
�a
�a
� I I���
� I� .T�
+ V�phep
����
oi x �
ng truc�la�
phe�
p d�
��
i h nh nen
� (C): gTam
�(C�
) : gTam
m I I���
� I�
�
gR 2
gR R 2
�
2 2
�a
�
�
+
Tam
�
I(
3;2)
+
Tam
�
I
=
�
[
I(
3;
2)]
(
; )
a
�
Vay
�: (C)
I���
� (C ) �
5 5
BK : R = 2
�
BK : R�
=R=2
�
2
2
� (C�
) : (x )2 (y )2 4
5
5
15 Trong mpOxy cho �
iem
� M(3; 5), ����
ng thang
� () : 3x + 2y 6 = 0, ����
ng tron
�(C) : (x+1)2 (y 2)2 9.
T�
m anh
� cua
�M, () va�
(C) qua phe�
p ���
oi x �
ng truc�(a) : 2x y + 1 = 0 .
HD :
�a
33 1
9 13
a) M(3; 5) I���
� M�
( ; ),(d) : x 2y 7 0,t�
iem
� H( ; )
5 5
5 5
4 15
b) + K= Ǯ (a) K( ; )
7 7
+ P �() : P(2;0) �K , Q = �a[P(2;0)] = ( 2;2)
� (�
) �(KQ) : x 18y 38 0
�a
9 8
9
8
c) + I(1; 2) I���
� I�
( ; ) , R�
=R=3
� (C�
) : (x + )2 (y )2 9
5 5
5
5
16 Cho �
iem
� M(2; 3), �
�
�
�
ng thang
� () : 2x + y 4 = 0, �
�
�
�
ng tron
�(C) : x 2 y 2 2x 4y 2 0.
T�
m anh
� cua
�M, () va�
(C) qua phe�
p ���
oi x �
ng qua Ox .
�Ox
�x x�
�x�
x
HD : Ta co�
: M(x;y) ���
� M�
(1) � �
(2)
�
y
�y y�
�y�
�
Ox � M�
gThay vao
�(2) : M(2; 3) ���
(2;3)
- 13 -
Bài tập về phép biến hình – Toán 11 nâng cao
gM(x;y) �() � 2x�
y�
4 = 0 � M���
(x ;y ) �(�
) : 2x y 4 = 0 .
2 y�
2 2x�
gM(x;y) �(C) : x2 y 2 2x 4y 2 0 � x�
4y �
2 0
� (x�
1)2 (y�
2)2 3 � M���
(x ;y ) �(C�
) : (x 1)2 (y 2)2 3
17 Trong mpOxy cho �
�
�
�
ng thang
� (a) : 2x y+3 = 0 . T�
m anh
� cua
�a qua �Ox .
�Ox
�
�
x�
x
x x�
Giai�: Ta co�
: M(x;y) I���
� M�
��
��
y y �
y y�
�
V�M(x;y) �(a) : 2x y+3 = 0 � 2(x�
) ( y �
)+3 = 0 � 2x�
y�
+3 = 0 � M�
(x��
; y ) �(a�
) : 2x y + 3 = 0
�
Oy
Vay
�: (a) I���� (a�
) : 2x y + 3 = 0
18 Trong mpOxy cho �
�
�
�
ng tron
�(C) : x 2 y 2 4y 5 = 0 . T�
m anh
� cua
�a qua �Oy .
�Oy
�
x�
x �
x x�
Giai�: Ta co�
: M(x;y) I���� M�
��
�
�
y
y
y
�
� y�
2 4(y �
2 y�
2 4y 5 = 0
V�M(x;y) �(C) : x 2 y2 4y 5 = 0 � ( x�
)2 y �
) 5 = 0 � x�
� M���
(x ; y ) �(C�
) : x2 y 2 4y 5 = 0
�
Oy
Vay
�: (C) I���� (C�
) : x 2 y 2 4y 5 = 0
19 Trong mpOxy cho �
thang
� (a) : 2x y 3 = 0 , ( ) : x 3y 11 = 0 , (C) : x2 y 2 10x 4y 27 = 0 .
a) Viet�bieu
�th�
�
c giai��
t ch cua
�phep
����
oi x �
ng truc��a .
b) T�
m anh
� cua
��
iem
� M(4; 1) qua �a .
c) T�
m anh
� : (�
) = �a ( ),(C�
) �a (C) .
Giai�
a) Tong
� quat�(a) : Ax + By + C=0 , A 2 B2 �0
uuuuu
r
uuuuu
r r
�a
r
Goi�M(x;y) I���
� M���
(x ; y ) , ta co�
: MM�
(x�
x; y�
y) c ung
� ph�
�
ng VTPT n = (A;B) � MM�
tn
x x�y y�
�
x�
x At �x�
x At
��
��
(t ��) . Goi�I la�
trung �
iem
� cua
�MM�
ne�
n I(
;
) �(a)
y�
y Bt �
y�
y Bt
2
2
�
x x�
y y�
x x At
y y Bt
� A(
) B(
) C 0 � A(
) B(
)C 0
2
2
2
2
2(Ax + By + C)
� (A 2 B2 )t 2(Ax + By + C) � t
A 2 B2
�
2A(Ax + By + C)
2B(Ax + By + C)
��
x�
x
; y�
y
�
A2 B2
A2 B2
�
�
4(2x y 3)
3
4
12
x�
x
x�
x y
�
�
�
�
5
5
5
5
Ap
�dung
� ket�qua�
tren
�ta co�
: �
��
2(2x
y
3)
4
3
6
�
�y�
y�
y
y y
5
5
5
�
� 5
�a
4 7
b) M(4; 1) I���
� M�
( ; )
5 5
�
a � �
c) I���
: 3x y 17 0
�
a � (C�
d) (C) I���
) : (x 1)2 (y 4)2 2
- 14 -
Bài tập về phép biến hình – Toán 11 nâng cao
20 Trong mpOxy cho ñöôøng thaúng () : x 5y 7 = 0 vaø (�
) : 5x y 13 = 0 . Tìm pheùp ñoái xöùng qua
truïc bieán () thaønh (�
).
Giaûi
1 5
Vì � � () vaø (�
) caét nhau . Do ñoù truïc ñoái xöùng (a) cuûa pheùp ñoái xöùng bieán () thaønh (�
) chính
5 1
laø ñöôøng phaân giaùc cuûa goùc taïo bôûi () vaø (�
).
�
x y 5 0 (a1)
��
x y 1 0 (a2 )
1 25
25 + 1
�
Vaäy coù 2 pheùp ñoái xöùng qua caùc truïc (1) : x y 5 0 , ( 2 ) : x y 1 0
Töø ñoù suy ra (a) :
| x 5y 7 |
| 5x y 13|
21 Qua phep
����
oi x �
ng truc��a :
1. Nh�
�
ng tam giac�nao
�bien
�thanh
� ch�
nh no�
?
2. Nh�
�
ng �
�
�
�
ng tron
�nao
�bien
�tha�
nh ch�
nh no�
?
HD :
1. Tam giac�co��
1 �
nh �truc�a , hai �
�
nh con
�lai��
oi��
x �
ng qua truc
�a .
2. ��
�
�
ng tron
�co��
tam �a .
22 T�
m anh
� cua
��
�
�
�
ng tron
�(C) : (x 1 )2 (y 2)2 4 qua phep
����
oi x �
ng truc�Oy.
PP : Dung
� bieu
�th�
�
c toa�
��
o � �S : (C�
) : (x 1)2 (y 2)2 4
23 Hai ABC va�
A�
B��
C cung
� nam
� trong mat�phang
� toa�
��
o va�
���
oi x �
ng nhau qua truc�Oy .
Biet�A( 1;5),B(4;6),C�
(3;1) . Hay
��
t m toa�
��
o cac��
�
nh A�
, B�
va�
C.
�S : A�
(1;5), B�
(4;6) va�
C( 3;1)
24 Xet�cac�h�
nh vuong
� , ngu�
giac��u
e�va��
luc giac���
eu . Cho biet�so�
truc
����
oi x �
ng t�
�
ng �
�
ng cua
�moi�
loai��
a giac���
eu ��
o va�
ch�ra cach
� ve�
cac�truc����
oi x �
ng ��
o.
- 15 -
Bài tập về phép biến hình – Toán 11 nâng cao
�S :
gH�
nh vuong
� co�
4 truc
����
oi x �
ng , ���
o la cac��
�
�
�
ng thang
��
i qua 2 �
�
nh ��
oi dien
�va�
cac��
�
�
�
ng thang
�
�
i qua trung �
iem
� cua
�cac�cap
�canh
� ��
oi dien
�.
gNgu�
giac���
eu co�
5 truc����
oi x �
ng ,���
o la cac��
�
�
�
ng thang
��
i qua �
�
nh ��
oi dien
�va��
tam cua
�ngu�
giac
���
eu
gLuc�giac���
eu co�
6 truc����
oi x �
ng , ���
o la cac��
�
�
�
ng thang
��
i qua 2 �
�
nh ��
oi dien
�va�
cac��
�
�
�
ng thang
��
i
qua trung �
iem
� cua
�cac�cap
�canh
� ��
oi dien
�.
25 Goi�d la�
phan
�giac�trong tai�A cua
�ABC , B�
la�
anh
� cua
�B qua phep
��oi��
x �
ng truc��d . Khang
��
�
nh
nao
�sau ��
ay sai ?
A. Neu
�AB < AC th�B�
��
tren
�canh
� AC .
B. B�
la�
trung �
iem
� canh
� AC .
C. Neu
�AB = AC th�B�
�C .
D. Neu
�B�
la�
trung �
iem
� canh
� AC th�AC = 2AB .
�S : Neu
�B�
= �d (B) th�B�
�AC .
gA ��
ung . V�AB < AC ma�
AB�
= AB nen
�AB�
< AC � B��
�
tren
�canh
� AC .
1
gB sai . V�gia�
thiet�bai�toan
�kho �
ng ��
u khang
��
�
nh AB = AC.
2
�
�
�
gC ��
ung . V�AB = AB ma�
AB = AC nen
�AB = AC
B C .
gD ��
ung . V�Neu
�B�
la�
trung �
iem
� ca�
nh AC th�AC=2AB�
ma�
AB�
=AB nen
�AC=2AB .
26 Cho 2 �
��
�
ng thang
� a va�
b cat�nhau tai�O . Xet�2 phep
����
oi x �
ng truc��a va�
�b :
�
�
a � B I���
b � C . Khang
A I���
��
�
nh nao
�sau ��
ay khong
� sai ?
A. A,B,C ���
�
�
ng tron
�(O, R = OC) .
B. T��
giac�OABC noi�tiep
�.
C. ABC can
���
B
D. ABC vuong
� ��
B
HD : gA. Khong
� sai . V�
d1 la�
trung tr��
c cua
�AB � OA = OB , d 2 la�
trung tr��
c
cua
�BC � OB = OC � OA = OB = OC � A,B,C ���
�
�
ng tron
�(O, R = OC) .
gCac�cau
�B,C,D co�
the �
sai .
27 Cho ABC co�
hai truc����
oi x �
ng . Khang
��
�
nh nao
�sau ��
ay ��
ung ?
A. ABC la�
vuong
�
B. ABC la�
vuong
� can
� C. ABC la�
��
eu
HD : G�
a s�
�
ABC co�
2truc����
oi x �
ng la�
AC va�
BC
�
AB = AC
��
� AB AB BC � ABC ��
eu .
BC = BA
�
� 110o. T�
� va�
� ��
28 Cho ABC co�
A
nh B
C
e ABC
co�
truc
����
oi x �
ng .
�
�
� = 45o va�
� 25o
o
A. B = 50 va�
C 20o
B. B
C
D. ABC la�
can
�.
� = 40o va�
� 30o
C. B
C
HD : Choïn D . Vì : ABC coù truïc ñoái xöùng khi ABC caân hoaëc ñeàu
� 110o 90o � ABC caân taïi A , khi ñoù :
Vì A
� 180o 110o
180o A
�
�
BC
35o
2
2
29 Trong cac�h�
nh sau , h�
nh nao
�co�
nhieu
�truc����
oi x �
ng nhat�?
A. H�
nh ch�
�
nhat�
B. H�
nh vuong
�
C. H�
nh thoi
�S : Chon
�B. V�: H�
nh vuong
� co�
4 truc����
oi x �
ng .
30 Trong cac�h�
nh sau , h�
nh nao
�co�
�
t truc����
oi x �
ng nhat�?
A. H�
nh ch�
�
nhat�
B. H�
nh vuong
�
C. H�
nh thoi
�S : Chon
�D. V�
: H�
nh thang can
�co�truc
1 ����
oi x �
ng .
- 16 -
�=C
� 35o
D. B
D. H�
nh thang can
�.
D. H�
nh thang can
�.
Bài tập về phép biến hình – Toán 11 nâng cao
31 Trong cac�h�
nh sau , h�
nh nao
�co�
3 truc����
oi x �
ng ?
A. H�
nh thoi
B. H�
nh vuong
�
C. ��
eu
D. vuon
�g can
�.
�S : Chon
�C. V�: ��
eu co�
3 truc��
oi��
x �
ng .
32 Trong cac�h�
nh sau , h�
nh nao
�co �
n hieu
�h�
n 4 truc����
oi x �
ng ?
A. H�
nh vuong
�
B. H�
nh thoi
C. H�
nh tron
�
�S : Chon
�C. V�
: H�
nh tron
�co�
vo�
so�
truc����
oi x �
ng .
D. H�
nh thang can
�.
33 Trong cac�h�
nh sau , h�
nh nao
�khon
�g co�
truc����
oi x �
ng ?
A. H�
nh b�
nh ha�
nh
B. ��
eu
C. can
�
D. H�
nh thoi .
�S : Chon
�A. V�: H�
nh b�
nh hanh
� kho�
ng co�
truc����
oi x �
ng .
34 Cho hai h�
nh vuo�
ng ABCD va�
AB���
C D co�
canh
� ��
eu bang
� a va�
co�
�
�
nh A chung .
Ch�
�
ng minh : Co�
the��
th �
c hien
�mot�phep
����
oi x �
ng truc�bien
�h�
nh vuon
�g ABCD thanh
��
AB���
CD .
HD : G�
a s�
�
: BC �B��
C =E.
�B
��
Ta co�
: AB = AB�
,B
90o,AE chung .
�ABE
���
= AB
�
F
�
�
EB = EB�
B I AE
�
�biet�AB = AB�
B�
�
�
EC = EC�
Mat�khac�: �
���
C I AE C�
AC = AC�
=a 2
�
� �
BAB
�
�
o
Ngoai�ra : AD�
= AD va�
D�
AE DAE 90
2
�A
�AE
����
D I ��
D� ABCD I
AB���
CD
35 Goïi H laø tröïc taâm ABC . CMR : Boán tam giaùc ABC , HBC , HAC , HAC coù
ñöôøng troøn ngoaïi tieáp baèng nhau .
HD :
� =C
� (cung
� )
Ta co�
:A
� chan
�cung BK
1
2
� =C
� (goc�co�
� =C
�
A
canh
� t�
�
ng �
�
ng ) � C
1
1
1
2
� CHK can
�� K ���
oi x �
ng v�
�
i H qua BC .
Xet�phep
����
oi x �
ng truc�BC .
�
�
�
BC H ; B I����
BC B ; C I����
BC C
Ta co�
: K I����
�
BC ��
Vay
�: ��
�
�
ng tron
�ngoai�tiep
�KBC I����
��
ng tron
�ngoai�tiep
�HBC
36 Cho ABC va�
�
�
�
�
ng thang
�a�
i qua �
�
nh A nh�
ng khong
��
i qua B,C .
a) T�
m anh
� ABC qua phep
����
oi x �
ng �a .
b) Goi�G la�
trong
� tam
� ABC , Xac��
�
nh G�
la�
anh
� cua
�G qua phep
����
oi x �
ng �a .
Giaûi
a) Vì a laø truïc cuûa pheùp ñoái xöùng Ña neân :
gA �a � A Ña (A) .
gB,C Ͼ�����
a neân Ña : B I
b) Vì G Ͼ��
a neân Ña : G I
B�
,C I
C�
sao cho a laø trung tröïc cuûa BB�
,CC�
G�
sao cho a laø trung tröïc cuûa GG�
.
- 17 -
Bài tập về phép biến hình – Toán 11 nâng cao
37 Cho �
�
�
�
ng thang
� a va�
hai �
iem
� A,B nam
� cung
� ph����
a oi v �
i a . T�
m tren
��
�
�
�
ng
thang
�a�
iem
� M sao cho MA+MB ngan
�nha�
t.
Giai�: Xet�phep
����
oi x �
ng �a : A I��
� A�
.
M �a th�MA = MA�
. Ta co�
: MA + MB = MA �
+ MB �A�
B
�e�
MA + MB ngan
�nhat�th�
chon
�M,A,B thang
� hang
�
Vay
�: M la�
giao �
iem
� cua
�a va�
A�
B.
38 (SGK-P13)) Cho goc�nhon
�xOy va�
M la�
mot��
iem
� ben
�trong goc���
o . Hay
�
t�
m�
iem
� A tren
�Ox va�
�
iem
� B tren
�Oy sao cho MBA co�
chu vi nho�
nhat�.
Giai�
Goi�N = �Ox (M) va�
P = �Ox (M) . Khi ��
o : AM=AN , BM=BP
T�
�
��
o : CVi = MA+AB+MB = NA+AB+BP �NP
(�
�
�
�
ng gap
�khuc���
�
�
�
ng tha�
ng )
MinCVi = NP Khi A,B lan
��
l �
�
t la�
giao �
iem
� cua
�NP v�
�
i Ox,Oy .
39 Cho ABC can
�tai�A v�
�
i�
�
�
�
ng cao AH . Biet�A va�
H co�
�
�
nh . T�
m tap
�h�
�
p
�
iem
� C trong moi�tr�
�
�
ng h�
�
p sau :
a) B di ��
ong tren
��
�
�
�
ng thang
�.
b) B di ��
ong tren
��
�
�
�
ng tro�
n tam
� I, ban
�k�
nh R .
Giai�
a) V�
: C = �AH (B) , ma�
B � nen
�C ��
v�
�
i �
= �AH ( )
Vay
�: Tap
�h�
�
p cac��
iem
� C la�
��
�
�
ng thang
� �
b) T�
�
ng t�
�
: Tap
�h�
�
p cac��
iem
� C la�
�
�
�
�
ng tron
�tam
� J , ban
�k�
nh R la�
anh
� cua
�
�
�
�
�
ng tron
�(I) qua �AH .
Vấn đề 4 : PHÉP ĐỐI XỨNG TÂM
1 �N : Phep
����
oi x �
ng tam
� I la�
mot�phep
�d�
��
i h nh bien
�moi��
iem
� M than
�h �
iem
� M�
���
oi x �
ng v�
�
i M qua I.
Phep
����
oi x �
ng qua mot��
iem
� con
�goi�la�
phep
���
oi tam
�.
�iem
� I goi�la�m
ta� cua
�cua
�phep
����
oi x �
ng hay �
�
n gian
�la�����
tam oi x �
ng .
uuur
uuu
r
Kí hieäu : ÑI (M) M�
� IM�
IM .
gNeu
�M �I th�M�
�I
gNeu
�M �I th�M�
�I (M) � I la�
trung tr��
c cua
�MM�
.
g�N :�iem
� I la�����
tam oi x �
ng cua�h�
nh H � �I (H) H.
Chu��
y : Mot��
h nh co�
the�
khong
� co�
tam
����
oi x �
ng .
�I
2 Bieu
�th�
�
c toa
���
o : Cho I(x o ; y o ) va�
phep
����
oi x �
ng tam
� I : M(x;y) I���
� M�
�I (M) (x��
; y ) th�
x�
= 2xo x
�
��
y 2yo y
�
3 T�
nh chat�:
1. Phep
����
oi x �
ng tam
� bao
�toan
�khoang
� cach
� gi�
�
a hai �
iem
� bat��
k .
2. Bien
�mot�tia thanh
� tia .
3. Bao
�toan
��
t nh thang
� hang
� va�
th�
�
t�
�
cua
�cac��
iem
� t�
�
ng �
�
ng .
4. Bien
�mot��
oan
�thang
� thanh
��
oan
�thang
� bang
� no�
.
5. Bien
�mot��
�
�
�
ng thang
� thanh
� mot��
�
�
�
ng thang
� song song hoac�trung
� v�
�
i�
�
�
�
ng thang
� ��
a cho .
6. Bien
�mot�goc�thanh
� goc�co�
so�
�
o bang
� no�
.
7. Bien
�tam giac�thanh
� tam giac�ban
�g no�
. ( Tr�
�
c tam
� � tr�
�
c tam
� , trong
� tam
� � trong
� tam
�)
- 18 -
Bài tập về phép biến hình – Toán 11 nâng cao
8. ��
�
�
ng tron
�thanh
��
�
�
�
ng tron
�ba�
ng no�
. ( Tam
� bien
�thanh
� tam
� : I I��
� I�
, R�
=R)
B . BÀI TẬP
1 T�
m anh
� cua
�cac��
iem
� sau qua phep
����
oi x �
ng tam
�I :
1) A( 2;3) , I(1;2)
� A�
(4;1)
�
2) B(3;1) , I( 1;2)
� B (5;3)
3) C(2;4) , I(3;1)
� C�
(4; 2)
Giaûi :
uur
uur
x�
1 3
x�
4
a) Gæa söû : A�
ÑI (A) � IA IA � (x�
1; y�
2) (3;1) �
�
� A�
(4;1)
y�
2 1
y�
1
Caùch �: Duøng bieåu thöùc toaï ñoä
2 T�
m anh
� cua
�cac��
�
�
�
ng thang
� sau qua phep
����
oi x �
ng tam
�I :
1) () : x 2y 5 0,I(2; 1)
� (�
) : x 2y 5 0
2) () : x 2y 3 0, I(1; 0)
� (�
) : x 2y 1 0
3) () : 3x 2y 1 0, I(2; 3)
� (�
) : 3x 2y 1 0
Giai�
PP : Co�
3 cach
�
Cach
� 1: Dung
� bieu
�th�
�
c toa�
��
o
�
Cach
� 2 : Xac
��
�
nh dang
� // , roi�dung
� cong
� th�
��
c t nh khoang
� cach
� d(;�
) � �
.
Cach
� 3 : Lay
�bat�ky�
A,B � , roi��
t m a�
nh A��
,B ��
� �
�A�
B�
�I
�
�
x�
4x
x 4 x�
1) Cach
� 1: Ta co�
: M(x;y) I���
� M�
��
��
y 2 y �
y 2 y�
�
V�M(x;y) � � x 2y 5 0 � (4 x�
) 2(2 y�
) 5 0 � x�
2y�
5 0
� M���
(x ;y ) ��
: x 2y 5 0
�I
Vay
�: () I���
� (�
) : x 2y 5 0
Cach
� 2 : Goi��
= �I () � �
song song � �
: x + 2y + m = 0 (m �5) .
|5|
|m|
�
m 5 (loai)
�
Theo ��
e : d(I;) = d(I;�
)�
� 5 | m | � �
m 5
�
12 22
12 22
� ( �
) : x 2y 5 0
�
�
�
��
Cach
� 3 : Lay
�: A( 5;0),B( 1; 2) � � A (9; 2),B (5; 0) � �A B : x 2y 5 0
3 T�
m anh
� cua
�cac
��
�
�
�
ng tron
�sau qua phep
����
oi x �
ng tam
�I :
1) (C) : x2 (y 2)2 1,E(2;1)
2) (C) : x2 y2 4x 2y 0,F(1; 0)
3) (P) : y = 2x2 x 3 , tam
� O(0;0) .
� (C�
) : (x 4)2 y2 1
� (C�
) : x2 y2 8x 2y 12 0
�/ nghia�
hay bieu
�th�
�
c toa�
��
o
��������������(P�
) : y = 2x 2 x 3
HD : a) Co�
2 cach
� giai�:
Cach
� 1: Dung
� bieu
�th�
�
c toa�
��
o.
�E
Cach
� 2 : T�
m tam
� I I���
� I�
,R�
R (��
a cho) .
b) T�
�
ng t�
�
.
4 Cho hai �
iem
� A va�
B .Cho biet�phep
�bien
���
oi M than
�h M�
sao cho AMBM�
la�
mot��
h nh b�
nh han
�h .
- 19 -
Bài tập về phép biến hình – Toán 11 nâng cao
HD :
uuuu
r uuuur
�
�
MA BM�
Neu
�AMBM�
la�
h�
nh b�
nh hanh
� � �uuur uuuur
MB AM�
uuuuu
r uuuu
r uuuur uuuu
r uuur �
V�
: MM�
MA AM�
MA MB (1)
uur
uur
Goi�I la�
trung
�
iem
�
cua
�
AB
.
Ta
co
�
:
IA
IB
uuuuu
r uuu
r uur uuu
r uur uuuuu
r
uuu
r
�
�
T�
�(1) �uMM
MI
IA
MI
IB
�
MM
2MI
uu
r uuur
� MI IM�
� M�
�I (M) .
5 Cho ba �
�
�
�
ng tron
�bang
� nhau (I1; R),(I2 ; R),(I3; R) t�
�
ng ��
oi tiep
�
xuc�nhau tai�A,B,C . G�
a s�
�
M la�
mo�
t�
iem
� tren
�(I1; R) , ngoai�ra :
�I
�C
�A
�B
1 �Q .
M I���
� N ; N I���
� P ; P I���
� Q . CMR : M I���
HD :
�Do (I1; R) tiep
�xuc�v�
�
i (I2 ; R) tai�A , nen
�:
uuuur
uuuur
�A
�A
�A
M I�����
N ; I1 I�������
I2 MI
1 I
NI2
MI1
NI2 (1)
�Do (I2 ; R) tiep
�xuc�v�
�
i (I3; R) tai�B , nen
�:
uuuur
uuur
�B
�B
�B
N I�����
P ; I2 I �������
I3 NI
2 I
PI3
NI 2
PI3 (2)
�Do (I3; R) tiep
�xuc�v�
�
i (I1; R) tai�C , nen
�:
uuur
uuur
�C
�
�C
P I��
�������
Q; I3 I C I1
PI3 I ��
QI1 � PI3 QI1 (3)
uuuur
uuur
T�
�
(1),(2),(3) suy ra : MI1 QI1 � M �I (Q) .
1
5 Cho ABC la�
tam giac�vuong
� tai�A . Ke�
�
�
�
�
ng cao AH . Ve�
ph�
a
ngoai�tam giac�hai h�
nh vuong
� ABDE va�
ACFG .
a) Ch�
�
ng minh tap
�h�
�
p6�
iem
� B,C,F,G,E,D co�
mot�truc����
oi x �
ng .
b) Goi�K la�
trung �
iem
� cua
�EG . Ch�
�
ng minh K �
�
tren
��
�
�
�
ng than
�g AH .
c) Goi�P = DE �FG . Ch�
�
ng minh P �
�en
tr ��
�
�
�
ng thang
� AH .
d) Ch�
�
ng minh : CD BP, BF CP .
e) Ch�
�
ng minh : AH,CD,BF ��
ong qui .
- 20 -
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