Chuyen de Phep Bien Hinh Trong Mp (1)

7/24/2019 Chuyen de Phep Bien Hinh Trong Mp (1)

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PHP DI H!NH V" PHP #NG D$NGT%&NG M'T PH(NG

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- 1 -


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J. KIN THL ON

- Ph/0 Ai,n h1nh.N: Php bin hnh l mt quy tc vi mi im M c m!t ph"n#$ %&c 'nh ()c mt im

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6 (>n# tJMn thnh (>n# tJMn bIn# n. E3Lm bin thnh tLm: $I I R R = G6 c thnh #c bIn# n.

O . O"I TP

x =2x 11 Trong mpOxy cho phep bien hnh f: M(x;y) M =f(M) = .

y =y ! Tm a"nh c#"a cac $ie%m &a# : a) '(1;2) b) ( 1;2) c) (2; *)+ia"i : a) ' =f(') =(1;,)

b) =

+

f() =( ;/)c) =f() =(!; 1)

x =2x y 12 Trong mpOxy cho phep bien hnh f : M(x;y) M =f(M) = .

y =x 2y ! Tm a"nh c#"a cac $ie%m &a# : a) '(2;1) b) ( 1;!) c) ( 2;

*)+ia"i : a) ' =f(') =(*;!)

b) =f() =( *; *)c) =f() =( ; )

! Trong mpOxy cho phep bien hnh f : M(x;y) M =f(M) =(!x;y) . 0ay copha"i a3phep 453i

hnh hay 6

hong 7

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1 1 2 2

1 1 1 1

2 2 2 2

+ia"i : 8ay hai $ie%m ba9 6 M(x ;y )(x ;y )

a?y : f 6hong pha"i a3phep 453i hnh .(> co1 &o$ie%m f 6hong ba"o 9oa3n 6hoa"ng cach) .

* Trong mpOxy cho 2 phep bien hnh :

a) f : M(x;y) M =f(M) =(y ; x-2) b) g : M(x;y) M =g(M) =( 2x ; y1) . @hep bien hnh na3o 9ren $ay a3phep 453i hnh 7 AB :

1 2 a) f a3phep 453i hnh b) g 6hong pha"i a3phep 453i hnh ( C x x 9h M M )

, Trong mpOxy cho 2 phep bien hnh :

a) f : M(x;y) M =f(M) =(y 1 ; x) b)

1

g : M(x;y) M =g(M) =( x ; !y ) . @hep bien hnh na3o 9ren $ay a3phep 453i hnh 7 +ia"i : a) f a3phep 453i hnh b) g 6hong pha"i a3phep 453i hnh ( C y y

29h M M )

+

/ Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = ( 2x;y 1) . Tm a"nh c#"a $D53ng 9haEng ( ) : x !y 2 = F G#a phep bien hnh f . +ia"i :ach 1: B#3ng bie%# 9hDc 9oaH $o?

=

= + = = + = + =

xx = 2x x

Ta cof : M(x;y) M =f(M) = 2y y 1 y y 1x

> M(x;y) ( ) ( ) !(y 1) 2 F x /y 2 F M (x;y ) ( ) : x /y 2 F2

ach 2: 8ay 2 $ie%m ba9 6 M ( ) : M .

M

= =

= =

( ) : M(2;F) M f(M) ( *;1)

( ) : ( 1; 1) f() (2;F)

= + = =

#####rI#a M ( *;1) x* y 1 ( ) (M ): @T9aJc ( ) : @TTI ( ):x /y 2 F/ 1>T@ : M (/; 1)

+ +

2 2

Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = (x !;y 1) . a) MK f a3phep 453i hnh .

b) Tm a"nh c#"a $D53ng 9ro3n () : (x 1) (y 2) = * . ( ) : (x

2 22) (y !) = *

+

L Trong mpOxy cho phep bien hnh f : M(x;y) M =f(M) =(x !;y 1) . a) MK f a3phep 453i hnh . b) Tm a"nh c#"a $D53ng 9haEng ( ) : x 2y , =F .

c) Tm a"nh c#"a $D53ng 9ro3n () : (x

2 21) (y 2) =2 .2 2x y

4 ) Tm a"nh c#"a eip () : =1 .! 2

- ! -


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+

+

1 1 2 2

1 1 1 1

2 2 2 2

+ia"i : a) 8ay hai $ie%m ba9 6 M(x ;y )(x ;y )

a?y : f a3phep 453i hnh .

= + = + = + + = + =

b) ach 1: B#3ng bie%# 9hDc 9oaH $o?x = x ! x x ! Ta cof : M(x;y) M = f(M) =y y 1 y y 1

> M(x;y) ( ) (x !) 2(y 1) , F x 2y * F M (x;y ) (

+ =) : x 2y * F

= =

= =

ach 2: 8ay 2 $ie%m ba9 6 M ( ) : M .

M ( ) : M(, ;F) M f(M) (2;1)

( ) : (! ; 1) f() (F;2)

= + = =

#####r

I#a M (2;1) x 2 y 1

( ) (M ): @T9aJc ( ) : @TTI( ):x 2y * F2 1>T@ : M ( 2;1)ach !: > f a3phep 453i hnh nen f bien $D53ng 9haEng ( ) 9ha3nh $D53ng 9haEng (

= = + + =

) NN ( ) .

8ay M ( ) : M(, ;F) M f(M) (2;1) > ( ) NN ( ) ( ):x 2y m =F (m ,) . Bo : ( ) M (2;1) m = * ( ):x 2y * Fc) ach 1: B#3ng bie%# 9hDc 9oaH $o?

= + = + = + + =

2 2 2 2

x =x ! x x ! Ta cof : M(x;y) M =f(M) =y y 1 y y 1

> M(x;y) () : (x 1) (y 2) =2 (x *) (y !) 2

M (x;y)

+ + = = + + =

+ +

2 2

f 2 2( ) : (x *) (y !) 2Tam ( 1;2) Tam =f ( 1;2)P ( *;!)

ach 2: () () () : (x *) (y !) 2< : K = 2 < : K =K = 2

= + = + =

4) B#3ng bie%# 9hDc 9oaH $o?x = x ! x x ! Ta cof : M(x;y) M = f(M) =y y 1 y y 1

2 2 2 2 2 2x y (x !) (y 1) (x !) (y 1)> M(x;y) () : = 1 = 1 M (x ;y ) ( ) : = 1

! 2 ! 2 ! 2

+

+

Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = (x 1;y 2) .

a) MK f a3phep 453i hnh .b) Tm a"nh c#"a $D53ng 9haEng ( ) : x 2y !

+

2 2

2

2 2 2

= F.

c) Tm a"nh c#"a $D53ng 9ro3n () : (x !) (y 1) = 2 .

4) Tm a"nh c#"a parabo (@) : y = *x .

0Q : b) x 2y 2 = F c) (x 2) (y 1) = 2 4) (y 2) = *(x 1)

1F Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = ( x;y) .


5/39


0Q : hoHn . > M Ca3f(M) $oi xDng nha# G#a 9r#Hc 9#ng &ai .

1 1 2 2

1 2

12 Trong mpOxy cho 2 phep bien hnh :

f : M(x;y) M = f (M) = (x 2 ; y *) ; f : M(x;y) M = f (M) = ( x ; y) . Tm 9oaH $o?a"nh c#"a '(*; 1) G#a f roSi f ngha a39

1 22 1

U

m f f (')P .

0Q : '(*; 1) ' (/; ,) ' ( / ; , ) .

x11 Trong mpOxy cho phep bien hnh f : M(x;y) M =f(M) =( ; !y) . i hnh bin im M thnh im M Q2 ch2 uMM = r .

= =rr r

g

< W hie?# : T hay T .n# tJMn: 9h?n# @iG

U 4+y M (A) M (A )

;U g(A) $D53ng 9haEng (A ) $D53ng 9haEng c#3ng phD5ng

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+ + g Tam Tam (A) () (A ) ( ) (caSn 9m ) .

b6 : K b6 : K =K

ach 2 : B#3ng bie%# 9hDc 9oHa $o?. Tm x 9heo x 9m y 9heo y roSi 9hay Ca3o bie%# 9hDc 9oHa $o?.

ach ! : 8ay hai $ie%m phan bie?9 : M (A) M (A )

O. O"I TP

= = = + = + = =

r

#####r rr

1 Trong mpOxy . Tm a"nh c#"a M c#"a $ie%m M(!; 2) G#a phep 9Rnh 9ien 9heo Cec95 # = (2;1) .+ia"i

x ! 2 x ,Theo $Rnh ngha 9a co: M = T (M) MM # (x !;y 2) (2;1)# y 2 1 y 1

rr

M (,; 1)

2 Tm a"nh cac $ie%m ch[ra G#a phep 9Rnh 9ien 9heo Cec95 # : a) '( 1;1) # = (!;1)

r

' (2;!) b) (2;1) # = ( !;2)

r

( 1;!)

c) (!; 2) # = ( 1;!) (2;1)

= =

r###r ####r

r r

! Trong mpOxy . Tm a"nh ' aSn D5H9 c#"a $ie%m '(2;!) (1;1) G#a phep 9Rnh 9ien 9heo Cec95 # =(!;1) .

TWnh $o?4a3i ' ' .+ia"i

Ta co: ' =T (') (,;*) =T ()# # = =

= = =

= = = =

###r ####r

r r rr r r

#####r #######rrr r

1 2

1 2

(*;2) ' =\'\ , ' =\' \ , .

* ho 2 Cec95 # ;# . +[a &D"M T (M)M T (M ). Tm C $e%M T (M) .1 2 1 # 2 # 1 2 C

+ia"i

Theo $eS: M T (M) MM # M T (M ) M M1 # 1 1 2 # 1 1 2

= = = = + = =

r

######r ######r #####r #######rr r r r r r rr

# .2

e# : M T (M) MM C C MM MM M M # # .>a?y : C # #2 C 2 2 1 1 2 1 2 1 2

r, 0D53ng 9haEng caJ9 Ox 9aHi '( 1;F) caJ9 Oy 9aHi (F;2) . AaXy Cie9 phD5ng 9rnh $D53ng 9haEng a3a"nh c#"a G#a phep 9Rnh 9ien 9heo Cec95 # = (2; 1) .

= = = = = + = = +

r r

gr #####rg

+ia"i > : ' T (') (1; 1) T () (2;1) .# #G#a ' (1; 1) x 1 9 MaZ9 6hac : T ( ) $i G#a ' . Bo $o: p99& :# y 1 29>T@ : ' =(1;2)

= =

r

r

/ 0D53ng 9haEng caJ9 Ox 9aHi '(1;F) caJ9 Oy 9aHi (F;!) . AaXy Cie9 phD5ng 9rnh $D53ng 9haEng a3a"nh c#"a G#a phep 9Rnh 9ien 9heo Cec95 # =( 1; 2) . +ia"i

> : ' T (') (F; 2) # = =

= = = +

r

gr #####rg

T () ( 1;1) .#G#a ' (F; 2) x 9 MaZ9 6hac : T ( ) $i G#a ' . Bo $o: p99& :# y 2 !9>T@ : ' =( 1;!)

+ = +

rr

TD5ng 9DH : a) : x 2y * =F # =(F ; !) : x 2y 2 F b) : !x y ! =F # =( 1 ; 2) + + = : !x y 2 F

- / -


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+ =

r

r

2 2L Tm a"nh c#"a $D53ng 9ro3n () : (x 1) (y 2) * G#a phep 9Rnh 9ien 9heo Cec95 # =(1; !) .+ia"i

x=x 1 x =x 1ie%# 9hDc 9oaH $o?c#"a phep 9Rnh 9ien T a3:# y=y ! y =y!

> + = + + = + + =

+ + =

2 2 2 2 2 2 : M(x;y) () : (x 1) (y 2) * x (y 1) * M (x;y ) () : x (y 1) *

2 2 >a?y : 'Vnh c#"a () a3( ) : x (y 1) *

+

+

Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = (x 1;y 2) . a) MK f a3phep 453i hnh .

b) Tm a"nh c#"a $D53ng 9haEng ( ) : x 2y !

+

2 2

2

2 2 2

= F.

c) Tm a"nh c#"a $D53ng 9ro3n () : (x !) (y 1) = 2 .

4) Tm a"nh c#"a parabo (@) : y = *x .

0Q : b) x 2y 2 = F c) (x 2) (y 1) = 2 4) (y 2) = *(x

1)

1F Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = ( x;y) . M Ca3f(M) $oi xDng nha# G#a 9 r#Hc 9#ng &ai .

+ + = +

r

r

2 2 Tm a"nh c#"a $D53ng 9ro3n () : (x !) (y 2) 1 G#a phep 9Rnh 9ien 9heo Cec95 # = ( 2;*) .x= x 2 x = x 2+ia"i : ie%# 9hDc 9oaH $o?c#"a phep 9Rnh 9ien T a3:# y= y * y = y *

+ + = + = + =

+ =

2 2 2 2 2 2> : M(x;y) () : (x !) (y 2) 1 (x 1) (y 2) 1 M (x;y) ( ) : (x 1) (y 2) 12 2

>a?y : 'Vnh c#"a () a3() : (x 1) (y 2) 1 + + = + =

+ + =

r

r

2 2 2 2T TD5ng 9DH : a) () : (x 2) (y !) 1 # =(!;1) ( ) : (x 1) (y 2) 12 2 b) () : x y 2x *y * F # =( 2;!) ( ) + + =2 2: x y 2x 2y F

g

1F Trong he?9r#Hc 9oaH $o?Oxy xac $Rnh 9oaH $o?cac $[nh Ca3B c#"a hnh bnh ha3nh 'B bie9 $[nh'( 2;F) $[nh ( 1;F) Ca3giao $ie%m cac $D53ng cheo a3(1;2) .+ia"i

+oHi (x;y) .Ta = = =

= = = = =

=

##r ##r ##r

g##r ##r

##r

g##r##r

co: (x 1;y 2)' (!;2) (2; 1)> a39r#ng $ie%m c#"a ' nen :

x 1 ! x * =T () ' (*;*)' y 2 2 y *> a39r#ng $ie%m c#"a ' nen :

B =T () B = =

= =

##r x 1 2 x !B B B(!;*)y 2 2 y *B B

a3i 9a?p 9D5ng 9DH : '( 1;F)(F;*)(1;1) (!;2)B(2; 2) .

11 Rh2 ; (>n# th"n# Q2n# Q2n# nhu dv d .


8/39

T#Cr r


=

=

#####r ###r###r####r ####r ###r

+ia"i : hoHn 2 $ie%m co$Rnh ' 4 ' 4

8ay $ie%m 9#y3yM 4 . +[a &D": M =T (M) MM '' M' M M N NM' M 4 4 =T (4)' ha?n xe9 : oCo&ophep 9Rnh

##r

9ien bien 4 9ha3nh 4 .

12 ho 2 $D53ng 9ro3n (K) Ca3( K ) .AaXy ch[ra mo?9 phep 9Rnh 9ien bien (K) 9ha3nh ( K ) .

+ia"i : 8ay $ie%m M 9#y3y9ren (K) . +[a &D": M =T (M) M =

= = =

#####r ##r###r ####r ##r

M

M M M M K M ( K ) ( K ) =T (K)P

1! ho hnh bnh ha3nh 'B hai $[nh ' co$Rnh 9am 9hay $o%i 4i $o?ng 9ren $D53ng 9ro3n () .Tm G#yX9Wch 9r#ng $ie%m M c#"a caHnh .+ia"i

+oHi ] a39r#ng $ie%m caHnh ' . a?y M a3a"nh c#"a G#a phep 9Rnh 9ien T . Q#y ra : I#yX9Wch c#"a M a3]a"nh c#"a $D53ng 9ro3n () 9rong phep 9Rnh 9ien 9heo Cec95 ]

r21* Trong he?9r#Hc 9oaH $o?Oxy cho parabo (@) : y =ax . +oHi T a3phep 9Rnh 9ien 9heo Cec95 # =(mn) Ca3(@ ) a3a"nh c#"a (@) G#a phep 9Rnh 9ien $o. AaXy Cie9 phD5ng 9rnh c#"a

=

r #####r #####rrg

#####r r

#

(@ ) .+ia"i :

T M(x;y) M (x;y) 9a co: MM = # C5i MM = (x x ; y y)

x x = m x = x m > MM = #y y = n y = y n

2 2Ma3:M(x;y) (@):y ax y n = a(x m) y =

+ +

+ + +

rrr r

2 2a(x m) n M (x ;y) (@ ) : y = a(x m) n2 2 2

>a?y : 'Vnh c#"a (@) G#a phep 9Rnh 9ien T a3(@) : y =a(x m) n y = ax 2amx am n .#1, ho $9 : /x 2y 1=F . Tm Cec95 # F $e% = T ( ) .#+i =

r r r rrr

a"i : >T@ c#"a a3a = (2; /) . 0e%: = T ( ) # c#3ng phD5ng a .


9/39


+ = = = = = = =

+ + =

r r

###r ####rr r

# #

+ia"iT T

'(!;F) +( 1;!) + (x;y )x 1 * x ,> '+ ( *;!) # . Theo $eS: ++ # + ( ,;/).y ! ! y /

2 21 Trong maZ9 phaEng Oxy cho 2 $D53ng 9ro3n () : (x 1) (y !) 2(

+ + + =

r

2 2 ):x y 1Fx *y 2, F.

ohay 6hong phep 9Rnh 9ien Cec95 # bien () 9ha3nh ( ) .AB : () co9am (1; !) ban 6Wnh K =2 ; ( ) co9am (,; 2) ban 6Wnh K=2 . Ta 9hay : K =

rK =2 nen cophep 9Rnh 9ien 9heo Cec95 # =(*;1) bien () 9ha3nh ( ) .

2F Trong he?9r#Hc 9oaH $o?Oxy cho hnh bnh ha3nh O' C5i '( 2;1) Ca3 :2x y , =F . Tm 9a?p h5Hp $[nh 7+ia"i

= = =

= = = = = +

r

###r ###r rrg###r r

g

g

#

> O' a3hnh bnh ha3nh nen : 'O (2; 1) T () C5i # =(2; 1)#T x x 2 x x 2

(x;y) (x;y ) . Bo : #y y 1 y y 1

(x;y) 2x y , =F 2x y 1F =F (x;

y ) :2x y 1F =F21 ho ' . +oHi ' aSn D5H9 a39r#ng $ie%m cac caHnh ''. +oHi O O O Ca3 1 1 1 1 2 ! 1 2 ! 9D5ng Dng a3cac 9am $D53ng 9ro3n ngoaHi 9iep Ca3cac 9am $D53ng 9ro3

=

###r

###r

1

'2

n no?i 9iep c#"a ba 9am giac ' 1 1 ' Ca3' . hDng minh raYng : O O O .1 1 1 1 1 2 ! 1 2 !AB :

_e9 phep 9Rnh 9ien : T bien ' ' .1 1 1 1'

2T

' ' ;O1 1 1 1

`

= =

= =

###r ###r

######r ####r

###r ###r

######r ####r ######r

1 1

' '2 2

T T

O ; .1 2 1 2 O O O O .1 2 1 2 1 2 1 28y#a?n 9D5ng 9DH : _e9 cac phep 9Rnh 9ien T T y ra :1 1

'2 2

O O Ca3O O 2 ! 2 ! ! 1 !

`

= = = ####r

O O O O O O O (c.c.c).1 2 ! 2 ! ! 1 ! 1 1 2 ! 1 2 !a a a

= = = =

= =###r

o o o

####r ###r

22 Trong 9Dgiac 'B co' = / !cm B 12cm ' /F 1,F Ca3B F . TWnh $o?4a3i cac caHnh Ca3B' .AB :

T _e9 : ' M 'M .Ta co: 'M a3hnh bnh ha3nh Ca3M !` a =o oF (C 1,F )

b = + + = =

= + = + =

o o o o o

o

o 8aHi co: B !/F (F /F 1,F ) /F MB !F . 0Rnh yha3m co& 9rong MB :

!2 2 2 2 2MB M B 2M.B.co&!F (/ !) (12) 2./ !.12. !/2

MB = /cm .1

Ta co: MB = B Ca3M = MB ! MB a39am giac2

= =

= = =

o oo

$eS#

MB a3nD"a 9am giac $eS# BM F Ca3MB' !F .

>a?y : MB' M'B M' !F 'MB a39am giac can 9aHi M .

- -


10/39


= =o/ !

BDHng M< 'B < a39r#ng $ie%m c#"a 'B n# th"n# nu l (>n# tJun# tJKc c 27n MM @hep $oi xDng G#a $D53ng 9haEng co3n goHi a3phep $oi xDng 9r#Hc . 0D53ng 9haEng a goHi a39r#Hc $oi xDng. 02 : @hep $oi xDng G#a $D53ng 9haEng a a3phep bien hnh bien mo

= = ######r ######r

a o o o

i $ie%m M 9ha3nh $ie%m M $oi xDngC5i M G#a $D53ng 9haEng a .

i hnh.g 1.@hep$oi xDng 9r#Hcbien ba $ie%m 9haEng ha3ng 9ha3nh ba $ie%m 9haEng ha3ng Ca3ba"o 9oa3n 9hD9DH c#"a cac

$ie%m 9D5ng Dng .2. 0D53ng 9haEng 9ha3nh $D53ng 9haEng .!.

AI :

Tia 9ha3nh 9ia . *. 0oaHn 9haEng 9ha3nh $oaHn 9haEng baYng no.

,. Tam giac 9ha3nh 9am giac baYng no. (TrDHc 9am 9rDHc 9am 9roHng 9am 9roHng 9am )

/. 0D53ng 9ro3n 9ha3nh $D53ng

9ro3n baYng no. (Tam bien 9ha3nh 9am : K =K )

. +oc 9ha3nh goc baYng no.

a@@:Tm a"nh M = 0 (M)1. (4) M 4 a2. A = 4 a!. A a39r#ng $ie%m c#"a MM M 7

a

a

@@ : Tm a"nh c#"a $D53ng 9haEng : = 0 ( ) TA1:( )NN(a) 1. 8ay ' ( ) : '

2. Tm a"nh ' = 0 (') !. ' NN(a)

`

- 1F -


11/39


a

TA2 : NN a1. Tm < = a

2. 8ay @ : @ < .Tm I = 0 (@) !. ( 1 =

+ = = =

gg

a

caJ9 a < a


12/39


=

gg

g

g

g

AB : a Ox =


13/39


g

gg

a8ay @ < I = 0 @( 1;!)P = (1; 1) . ( 8a3m 9D5ng 9DH nhD ca# a) )

I#a @( 1;!) +oHi $D53ng 9haEng (b) :

a + +

= + = +

= + = +

@ I I

@ I I

(b) (a) (b) : 2x y m = F . > (b) @( 1;!) m = 1 (b) :2x y 1 = F = (b) (a) (F;1) a39r#ng $ie%m c#"a @I

1 1x (x x ) F ( 1 x ) x2 2 1 1

y (y y ) 1 (! y )2 2

= =

= == =

g ###rg

I

I

1 I(1; 1)

y 1

I#a a?y : () < :

{

= +

+ + =

0aa

2 2

2 2Tam =0 ( !; 2)P ( ; )

( ) , ,K = 2 < : K=K =22 2

( ) : (x ) (y ) *, ,

+ =

2 21, Trong mpOxy cho $ie%m M(!; ,) $D53ng 9haEng ( ) : !x 2y / =F $D53ng 9ro3n () : (x1) (y 2) . Tm a"nh c#"a M ( ) Ca3() G#a phep $oi xDng 9r#Hc (a) : 2x y 1 =F .AB :

a) M(!; ,) + + =

+ =

0a

a

!! 1 1!M ( ; )(4):x 2y F9$ie%m A( ; )

, , , ,* 1,

b)


14/39


+

= = = = + + +

2 2Oy

0Oy

2 2 2 2 2

1L Trong mpOxy cho $D53ng 9ro3n () : x y *y , = F . Tm a"nh c#"a a G#a 0 .

x x x x+ia"i : Ta co: M(x;y) My y y y

> M(x;y) () : x y *y , = F ( x ) y *(y ) , = F x

+

+

2

2 2

0Oy 2 2

y *y , = F

M (x;y ) ( ) : x y *y , = F

>a?y : () ( ) : x y *y , = F

+ + +

2 2

a

a

1 Trong mpOxy cho $9haEng (a) : 2x y ! = F ( ) : x !y 11 = F () : x y 1Fx *y 2 = F . a) >ie9 bie%# 9hDc gia"i 9Wch c#"a phep $oi xDng 9r#Hc 0 . b) Tm a"nh c#"a $ie%m M(*; 1) G#a 0 .

=

+

= =

#####r #####rr r

a a

2 2

0

a

c) Tm a"nh : ( ) = 0 ( )( ) 0 () .+ia"i

a) To%ng G#a9 (a) : 'x y =F ' F

+oHi M(x;y) M (x;y) 9a co: MM (x x;y y) c#3ng phD5ng >T@T n = (';) MM 9nx

+ + = = + = = +

+ + + + + + + + = + + =

+ = =

2 2

x x y yx '9 x x '9( 9 ) . +oHi a39r#ng $ie%m c#"a MM nen ( ; ) (a)

y y 9 y y 9 2 2x x y y x x '9 y y 9

'( ) ( ) F '( ) ( ) F2 2 2 2

2('x y ) (' )9 2('x y ) 9

' + = = + +

= = + + = + = +

2 2

2 2 2 2

0a

2'('x y ) 2('x y )

x x ;y y' '

*(2x y !) ! * 12x x x x y, , , ,'p 4#Hng 6e9 G#a"9ren 9a co:

2(2x y !) * ! /y y y y y

, , , ,*

b) M(*; 1) M ( ;,

+ =

+ =

0a

0 2 2a

),

c) :!x y 1 F

4) () ( ):(x 1) (y *) 2

2F Trong mpOxy cho $D53ng 9haEng ( ) : x ,y = F Ca3( ) : ,x y 1! = F . Tm phep $oi xDng G#a 9r#Hc bien ( ) 9ha3nh ( ) .

+

+ia"i1 ,

> ( ) Ca3( ) caJ9 nha# . Bo $o9r#Hc $oi xDng (a) c#"a phep $oi xDng bien ( ) 9ha3nh ( ) chWnh, 1

a3$D53ng phan giac c#"a goc 9aHo b5"i ( ) Ca3( ) .

1

2

1 2

x y , F (a )\ x ,y \ \,x y 1!\ TD3$oy ra (a) :

x y 1 F (a )1 2, 2, 1 >a?y co2 phep $oi xDng G#a cac 9r#Hc ( ) : x y , F ( ):x y 1 F

+ = + = =+

+ = =

a

21 I#a phep $oi xDng 9r#Hc 0 : 1. hDXng 9am giac na3o bien 9ha3nh chWnh no7 2. hDXng $D53ng 9ro3n na3o bien 9ha3nh chWnh no7

- 1* -


15/39


+ =2 2

AB :1. Tam giac co1 $[nh 9r#Hc a hai $[nh co3n aHi $oi xDng G#a 9r#Hc a .

2. 0D53ng 9ro3n co9am a .

22 Tm a"nh c#"a $D53ng 9ro3n () : (x 1) (y 2) * G#a phep $oi xDng 9r#Hc Oy.

+ + =

2 2 @@ : B#3ng bie%# 9hDc 9oaH $o? 0Q : ( ) : (x 1) (y 2) *

2! Aai ' Ca3 ' c#3ng naYm 9rong maZ9 phaEng 9oaH $o?Ca3$oi xDng nha# G#a 9r#Hc Oy .ie9 '( 1;,)( *;/) (!;1) . AaXy

9m 9oaH $o?cac $[nh ' Ca3 .0Q : ' (1;,) (*;/) Ca3( !;1)

2* _e9 cac hnh C#ong ng#Xgiac $eS# Ca3#Hc giac $eS# . ho bie9 &o9r#Hc $oi xDng 9D5ng Dng c#"a moi oaHi $a giac $eS# $oCa3ch[ra cach CeXcac 9r#Hc $oi xDng $o.

g

g

0Q :Anh C#ong co* 9r#Hc $oi xDng $oa3cac $D53ng 9haEng $i G#a 2 $[nh $oi 4ie?n Ca3cac $D53ng 9haEng$i G#a 9r#ng $ie%m c#"a cac caZp caHnh $oi 4ie?n .

g#Xgiac $eS# co

g

, 9r#Hc $oi xDng $oa3cac $D53ng 9haEng $i G#a $[nh $oi 4ie?n Ca39am c#"a ng#Xgiac $eS# .

8#Hc giac $eS# co/ 9r#Hc $oi xDng $oa3cac $D53ng 9haEng $i G#a 2 $[nh $oi 4ie?n Ca3cac $D53ng 9haEng $i G#a 9r#ng $ie%m c#"a cac caZp caHnh $oi 4ie?n .

42, +oHi 4 a3phan giac 9rong 9aHi ' c#"a ' a3a"nh c#"a G#a phep $oi xDng 9r#Hc 0 . ' ' ma3' = ' nen ' ' 5"9ren caHnh ' .1 &ai . > gia"9hie9 ba3i 9oan 6hong $#"6haEng $Rnh ' = '.

2 $#ng . > ' = ' ma3' = ' nen ' = ' .

g

a b0 0a b

B $#ng . > e# a39r#ng $ie%m caHnh ' 9h '=2' ma3' =' nen '=2' .

2/ ho 2 $D53ng 9haEng a Ca3b caJ9 nha# 9aHi O . _e9 2 phep $oi xDng 9r#Hc 0 Ca30 :

'

.


16/39


a a a

a a a a a a a a

=

= = = =

o

o o o o o o o

2L ho ' co' 11F . TWnh Ca3 $e% ' co9r#Hc $oi xDng .

'. =,F Ca3 2F . =*, Ca3 2, . =*F Ca3 !F B. = !,

a

a a a

o o

o o o o

AB : hoHn B . > : ' co9r#Hc $oi xDng 6hi ' can hoaZc $eS#

> ' 11F F ' can 9aHi ' 6hi $o:

1LF ' 1LF 11F !,2 2

= >

= = = =

2 Trong cac hnh &a# hnh na3o conhieS# 9r#Hc $oi xDng nha9 7 '. Anh chDXnha?9 . Anh C#ong . Anh 9hoi B. Anh 9hang can . 0Q : hoHn . > : Anh C#ong co* 9r#Hc $oi xDng .

!F Trong cac hnh &a# hnh na3o coW9 9r#Hc $oi xDng nha9 7 '. Anh chDXnha?9 . Anh C#ong . Anh 9hoi B. Anh 9hang can . 0Q : hoHn B. > : Anh 9hang can co1 9r#Hc $oi xDng .

!1 Trong cac hnh &a# hnh na3o co! 9r#Hc $oi xDng 7 '. Anh 9hoi . Anh C#ong . $eS# B. C#ong can .

0Q : hoHn . > : $eS# co! 9r#Hc $oi xDng .

!2 Trong cac hnh &a# hnh na3o conhieS# h5n * 9r#Hc $oi xDng 7 '. Anh C#ong . Anh 9hoi . Anh 9ro3n B. Anh 9hang can . 0Q : hoHn . > : Anh 9ro3n coCo&o9r#Hc $oi xDng .

!! Trong cac hnh &a# hnh na3o 6hong co9r#Hc $oi xDng 7 '. Anh bnh ha3 nh . $eS# . can B. Anh 9hoi . 0Q : hoHn '. > : Anh bnh ha3nh 6hong co9r#Hc $oi xDng .

!* ho hai hnh C#o

ng 'B Ca3' B cocaHnh $eS# baYng a Ca3co$[nh ' ch#ng .hDng minh : o9he%9hDHc hie?n mo?9 phep $oi xDng 9r#Hc bien hnh C#ong 'B 9ha3nh3' B .AB : +[a &D": = .

a a = =

o

0'

Ta co: ' =' F ' ch#ng . =

' = ' bie9 ' ='

b

= =

o

0'

0 0' '

= MaZ9 6hac :

' = ' = a 2

'goa3i ra : 'B = 'B Ca3B ' B' F

2B B 'B ' B

!, +oHi A a39rDHc 9am ' . MK : on 9am giac ' A A' A' co $D53ng 9ro3n ngoaHi 9iep baYng nha# .

i i j

i i i i

1 2

1 1 1 2

0 0

AB :

Ta co: ' = (c#3ng chaJn c#ng < )

' = (goc cocaHnh 9D5ng Dng ) = A< can < $oi xDng C5i A G#a .

_e9 phep $oi xDng 9r#Hc .

Ta co: < A ; ;

0

0

>a?y : 0D53ng 9ro3n ngoaHi 9iep


17/39


a

!/ ho ' Ca3$D53ng 9haEng a $i G#a $[nh ' nhDng 6hong $i G#a . a) Tm a"nh ' G#a phep $oi xDng 0 . b) +oHi + a39roHng 9am ' _ac $Rnh + a3a"nh c#"a + G#a phep $oi xDng 0a.

a

aa

a

+ia"ia) > a a39r#Hc c#"a phep $oi xDng 0 nen:

' a ' 0 (') . a nen 0 : &ao cho a a39r#ng 9rDHc c#"a

b) > + a nen 0 :+ + &ao cho a a39r#ng 9rDHc

=

gg

c#"a ++ .

! ho $D53ng 9haEng a Ca3hai $ie%m ' naYm c#3ng phWa $oi C5i a . Tm 9ren $D53ng9haEng a $ie%m M &ao cho M'M ngaJn nha9 .

+ia"i : _e9 phep $oi xDng 0 :' ' .aM a 9h M' = M' . Ta c

o: M' M = M' M ' 0e%M' M ngaJn nha9 9h choHn M' 9haEng ha3ng

>a?y : M a3giao $ie%m c#"a a Ca3' .

!L (Q+i = @


18/39


= =

=

02 ie%# 9hDc 9oHa $o?: ho (x ;y ) Ca3phep $oi xDng 9am : M(x;y) M 0 (M) (x;y ) 9ho o x= 2x xoy 2y yo

! TWnh cha9 :

1. @hep $oi xDng 9am ba"o 9oa3n 6hoa"ng cach giD

Xa hai $ie%m ba9 6 . 2. ien mo?9 9ia 9ha3nh 9ia . !. a"o 9oa3n 9Wnh 9haEng ha3ng Ca39hD9DH c#"a cac $ie%m 9D5ng Dng .

*. ien mo?9 $oaHn 9haEng 9ha3nh $oaHn 9haEng baYng no.,. ien mo?9 $D53ng 9haEng 9ha3nh mo?9 $D53ng 9haEng &ong &ong hoaZc 9r#3ng C5i $D53ng 9haEng $aXcho ./. ien mo?9 goc 9ha3nh goc co

&o$o baYng no.

. ien9am giac 9ha3nh 9am giac baYng no. ( TrDHc 9am 9rDHc 9am 9roHng 9am 9roHng 9am )

L. 0D53ng 9ro3n 9ha3nh $D53ng 9ro3n baYng no. ( Tam bien 9ha3nh 9am : K =K )

O . O"I TP

1 Tm a"nh c#"a cac $ie%m &a# G#a phep $oi xDng 9am : 1) '( 2;!) (1;2) ' (*;1)

2) (!;1) ( 1;2) ( ,;!) !) (2;*) (!;1) (*; 2)

{ {+ia"i :

x 1 ! x *a) +[a &D": ' 0 (') ' ' (x 1;y 2) ( !;1) ' (*;1) y 2 1 y 1 ach : B#3ng bie%# 9hDc 9oaH $o?

= = = = = = =

##r ##r

+ + = + =

2 Tm a"nh c#"a cac $D53ng 9haEng &a# G#a phep $oi xDng 9am :

1) ( ):x 2y , F(2; 1) ( ):x 2y , F2) ( ) = + =

+ = :x 2y ! F(1;F) ( ):x 2y 1 F

!) ( ):!x 2y 1 F(2; !) + + = ( ):!x 2y 1 F

+ia"i@@ : o! cachach 1:B#3ng bie%# 9hDc 9oaH $o?ach 2: _ac $Rnh 4aHng NN roSi 4#3ng cong 9hDc 9Wnh 6hoa"ng cach 4( ; ) .ach !: 8ay ba9 6y3' roSi 9m a"nh '

= =

= =

' 0 x * x x * x

1) ach 1:Ta co: M(x;y) M

y 2 y y 2 y

+ + = + + = + = + =

+ =

> M(x;y) x 2y , F (* x ) 2( 2 y ) , F x 2y , F M (x ;y ) :x 2y , F

0 >a?y : ( ) ( ) :x 2y , Fach 2: +oHi = 0 ( ) &ong &ong

= = = = + +

: x 2y m = F (m ,) .

\,\ \ m \ m , (oaHi) Theo $eS: 4(; ) = 4(; ) , \ m\

m ,2 2 2 21 2 1 2 + =

+ = ( ):x 2y , F

ach !: 8ay : '( ,;F)( 1; 2) ' (; 2) (,;F) ' :x 2y , F

- 1L -


19/39


+ = + =

! Tm a"nh c#"a cac $D53ng 9ro3n &a# G#a phep $oi xDng 9am :2 2 2 21) ():x (y 2) 1(2;1) ( ):(x *) y 122) ():x + + + = + + =

+

2 2 2y *x 2y F(1;F) ( ):x y Lx 2y 12 F$ N nghiaXhay bie%# 9hDc 9oaH $o?2!) (@) : y =2x x ! 9am O(F;F) .

= =

2(@ ):y = 2x x !

AB:1) o 2 cach gia"i :ach 1: B#3ng bie%# 9hDc 9oaH $o?.0

ach 2: Tm 9am K K ($a Xcho) .2) TD5ng 9DH .

* ho hai $ie%m ' Ca3 .ho bie9 phep bien $o%i M 9ha3n

h M &ao cho 'MM a3mo?9 hnh bnh ha3nh .

= =

= + = +

= =

####r ####r

###r ####r

#####r ####r ####r ####r ###r

##r ##r#####r # r

AB :

M' Me# 'MM a3hnh bnh ha3nh

M 'M

> :MM M' 'M M' M (1)

+oHi a39r#ng $ie%m c#"a ' . Ta co: '

TD3 (1) MM M + + + = = =

## ##r ###r ##r #####r ###r

###r ###r ' M MM 2M

M M M 0 (M) ., ho ba $D53ng 9ro3n baYng nha# ( ;K)( ;K)( ;K) 9D3ng $oi 9iep1 2 !x#c nha# 9aHi ' . +[a &D"M a3mo?9 $ie%m 9ren

' 1

( ;K) ngoa3i ra :1000 0

M ; @ ; @ I . MK : M I .

=

####r ####r' ' '

AB :

Bo ( ;K) 9iep x#c C5i ( ;K) 9aHi ' nen :1 20 0 0 M ; M M (1)1 2 1 2 1 2

=

####r ###r

Bo ( ;K) 9iep x#c C5i ( ;K) 9aHi nen :2 !0 0 0

@ ; @ @ (2)2 ! 2 ! 2 !Bo ( ;K) 9iep x#c C5i ( ;K) 9aHi nen :! 1

0 0 0 @ I ; @! 1 !

=

= =

###r ###r

####r ###r

1

I @ I (!)1 ! 1 TD3(1)(2)(!) y ra : M I M 0 (I) .1 1

, ho ' a39am giac C#ong 9aHi ' .


20/39


b b= =

o o

B B B B

B

AB :

a) Bo : 'B *, Ca3' *, nen ba $ie%m B' 9haEng ha3ng .0 0 0 0

Ta co: ' ' ; B B ; ; + ;0

(TWnh cha9 hnh C#ong ).

>a?y : Ta?p

{ } b bb b

=

=

h5Hp / $ie%m +B co 9r#Hc $oi xDng chWnh a3$D53ng 9haEng B' .b) I#a phep $oi xDng 9r#Hc B' 9a co: ' = '+ nen ' '+.

hDng : ' '+ ( 2 $oi xDng =)

b i i i= = '+ ' (4o


21/39


b

b

=

=

+ =

= =o o

+oHi O =a b 9a co: OM =OM Ca3MOM 2'OM1 1 1 OM =OM Ca3M OM 2M O1 2 1 2 1Q#y ra : OM =OM Ca3MOM M OM 2('OM M O)2 1 1 2 1 1

hay MOM 2 F 1LF1

>a?y : O a= 39r#ng $ie%m c#"a M Ca3M .2Bo $o: M 0 (M) M (A)M (A) O a39am $oi xDng c#"a (A) .2 O 2 b

=

= = = =

o

o o

L ho ' co'M Ca3 a3cac 9r#ng 9#yen . MK : e# 'M =!F 9h ' $eS# .AB :

TDgiac 'M co'M M !F nen no?i 9iep $9ro3n 9am O b6Wnh K=' Ca3MO 2'M /F .0

_e9 : '

= = =

+ = + = =

o

M M

0 (O) (O ) 9h (O ) C ' (O) .1 1

0 0 (O) (O ) 9h (O ) C (O) .2 2

OO OO 2K1 2 O O K K 2K O O nen a39r#n1 2 1 2

;

g $ie%m O O .1 2Q#y ra : ' OO O (> c#3ng $oSng 4aHng C5i M) .1 2> OO O a39am giac $eS# nen ' a39am giac $eS# .1 2

Vn 5 PHP ?UJYJ. KIN THL ON

1 0 : Trong maZ9 phaEng cho mo?9 $ie%m O co$Rnh Ca3goc D5Hng giac . @hep bien hnh bien moi $ie%m

M 9ha3nh $ie%m M &ao cho OM =OM Ca3(OM;OM ) = $D5Hc goHi a3phep G#ay 9am O C5i

g

g

@hep G#ay hoa3n 9oa3n xac $Rnh 6hi bie9 9am Ca3goc G#ay


22/39


(O; )

N1 Trong maZ9 phaEng Oxy cho $ie%m M(x;y) . Tm M = I (M) .(O; )AB :

x = rco&+oHi M(x;y) . 0aZ9 : OM = r goc D5Hng giac (Ox;OM) = 9h My = r&in

I N N> : M M . +oHi M (x ;y) 9h $o

= + = +

+

N N?4a3i OM = r Ca3(Ox;OM ) = .Ta co:

x = rco&( ) = aco& .co& a&in .&in xco& y&in . y = r&in( ) = a&in .co& aco& .&in x&in yco& .

x = xco& y&inN >a?y : My= x&in yco&

+ +

+

(O; )

( ; )o o

( ; )

o o

0aZc bie?9 :I x = xco& y&inNN M M

y = x&in yco&

I x x = (x x )co& (y y )&inN o o o M My y = (x x )&in (y y )co&(x ;y ) o o o

I M

(x ;y )

`

`

`

+

x x = (x x )co& (y y )&inNN o o oMy y = (x x )&in (y y )co&o o o

o

o(O;*, )

2 Trong mpOxy cho phep G#ay I . Tm a"nh c#"a :(O;*, )

2 2 a) 0ie%m M(2;2) b) 0D53ng 9ro3n () : (x 1) y = *I

N N N+ia"i . +oHi : M(x;y) M (x ;y ) . Ta co: OM = 2 2 (Ox; OM)

= =

= + = +

o o o o o

o o o o o

=

x = rco&( *, ) rco& .co&*, r&in .&in*, x.co&*, y.&in*,NTh My = r&in( *, ) r&in .co&*, rco& .&in*, y.co&*, x.&in*,

+

2 2x= x y

N 2 2M2 2

y= x y2 2

o

o

o

g gg g

(O; *, )

(O;*, )

(O;*, )

INa) '(2;2) ' (F ;2 2)

I NTam (1;F) Tam 7b) > () : ( ) :6 : K = 2 6 : K = K = 2

I2 2 2 2N 2 2 (1;F) ( ; ) . >a?y : ( ) : (x ) (y ) =2 2 2 2

*

+

1 !x= x y

2 2! Trong mpOxy cho phep bien hnh f : . Ao"i f a3phep g 7! 1

y= x y2 2

- 22 -


23/39


+

+ia"i

x= xco& y&in! !Ta cof : M(x;y) M (x;y ) C5i f a3phep G#ay I

(O; )y= x&in yco& !! !

* Trong mpOxy cho $D53ng 9haEng ( ) : 2x y1= F . Tm a"nh c#"a $D53ng 9haEng G#a :

a) @hep $oi xDng 9am (1; 2). b) @hep G#ay I .(O;F )+ia"i

a) Ta co: M (x ;y ) = 0 (M) 9h bie%# 9hDc

o

x 2 x x 2 x9oHa $o?My * y y * y

> M(x;y) ( ) : 2x y1= F 2(2 x ) ( * y ) 1 F 2x y F M (x;y ) ( ):2x y F

= = = = + = + + =

=

(O;F )

0>a?y : ( ) ( ) : 2x y F

Ib) ach 1 : +oHi M(x;y) M (x ;y ) . 0aZ9 (Ox ; OM) = OM = r

Ta co(Ox ; OM ) = F OM r .

x = rco& M(x;y) ( ) : 2(y) ( x ) 1 = F x 2y 1 = F M (x;y ) ( ):x 2y 1 F

I >a?y : ( )

= + = = = = = + = = + + + =

o o

o

O;F ) ( ):x 2y 1 F + + =o

+ + =

o

o

o

(O;F )

(O;F )

(O;F )

I ach 2 : 8ay : M(F;1) ( ) M ( 1;F) ( )

I1 1 ( ;F) ( ) (F; ) ( )2 2

I ( ) ( ) M :x 2y 1 F

= =

o

o

g

g

(O;F )

(O;F )

I1

ach ! : > ( ) ( ) ( ) ( ) ma3he?&ogoc : 6 2 62

I M(F;1) ( ) M (1;F) ( )

I#a M (1;F)

( ) : ( )1h&g ; 6 = 2

+ + =: x 2y 1 F

, Trong maZ9 phaEng 9oaH $o?Oxy cho '(!;*) . AaXy 9m 9oaH $o?$ie%m ' a3a"nho c#"a ' G#a phep G#ay 9am O goc F .

AB:+oHi (!;F)(F;*) aSn D5H9 a3hnh chie# c#"a ' en cac 9r#Hc Ox

Oy . @hepoG#ay 9am O goc F bien hnh chDXnha?9 O' 9ha3nh hnh chDXnha?9 O ' .


24/39


= == = =

###r ###r###r ###r

/ Trong maZ9 phaEng 9oaH $o?Oxy . Tm phep G#ay I bien $ie%m '( 1;,) 9ha3nh $ie%m (,;1) .

O' O 2/AB : Ta co: O' ( 1;,) Ca3O (,;1)O'.O F O' O

= I( o

(') .O ; F )

=

= + = + =

o

o

####r ###ro

Trong maZ9 phaEng 9oaH $o?Oxy cho $ie%m M(*;1) . Tm = I (M) .(O ; F )

AB :

> =I (M) (OM;O) F OM.O = F *xy =F y= *x (1)(O ; F )

2 2Bo : OM O x y 1/ 1 1 (2) .+ia"i (1) Ca3

= o(2) 9a co: (1; *) hay ( 1;*) .

ThD"aHi : 0ieS# 6ie?n (OM;O) F 9a 9hay ( 1;*) 9hoa"maXn .`

=

= >

+ = = =

o

oL a)Trong maZ9 phaEng 9oaH $o?Oxy cho $ie%m '(F;!) . Tm = I (') .(O ; *, )AB : @hep G#ay I bien $ie%m ' Oy 9ha3nh $ie%m $9:y x9a co:

(O ; *, )x y F 2 2 . Ma3 O = x y ! x O' O !

=

+o

! ! !( ; ).

2 2 2* ! ! ! * !

b) ho '(*;!) . Tm = I (') ( ; )(O;/F ) 2 2

+ =

= + + =

+ =

o

o

2 2 ho $D53ng 9ro3n () : (x !) (y 2) * . Tm ( ) = I () .(O ; F )

2 2AB : Tm a"nh c#"a 9am : I () ( 2;!) ( ):(x 2) (y !) * .(O ; F )2 21F ho $D53ng 9ro3n () : (x 2) (y 2 !) , . Tm ( ) =

= + + =

o

o

I () .(O ; /F )

2 2AB : Tm a"nh c#"a 9am : I () ( 2;2 !) ( ):(x 2) (y 2 !) , .(O ; /F )

+ =

= + + + =

o

o

2 211 ho $D53ng 9ro3n () : (x 2) (y 2) ! . Tm ( ) = I () .(O ; *, )

2 2AB : Tm a"nh c#"a 9am : I () (1 2;1 2) ( ):(x 1 2) (y 1 2) ! .(O ; *, )

o

o

12 -@1P Trong maZ9 phaEng 9oaH $o?Oxy cho $ie%m '(2;F) Ca3$D53ng 9haEng (4) : x y 2 =F.Tm a"nh c#"a ' Ca3(4) G#a phep G#ay I .

(O ; F )AB :Ta co: '(2;F) Ox . +oHi =I (

(O ; F )`

+ = +

o o

o

') 9h Oy Ca3O' =O .

> 9oaH $o?' 9hoa"maXn p9 (4) : x y 2 =F nen ' (4) .Bo =I (') Ca39D5ng 9DH I (') =( 2;F)

(O ; F ) (O ; F )x y x y

nen I (4) = () : 1(O ; F ) x y 2 2

`

= + =1 x y 2 F

- 2* -


25/39


+ =

+

o

o

1! ho (4) : x !y 1 =F . Tm = I (4) . ( ) : !x y 1 F(O ; F )

1* ho (4) : 2x y 2 = F . Tm =I (4) .(O ; /F )

1 !a"nhAB : 4 Ox ='(1;F) 4 Oy =(F;2) ' ( ; ) ( !;1)

2 2

( ) : ( ! 2 + + =)x (2 ! 1)y * F

o

o

1, ho 9am giac $eS# ' co9am O Ca3phep G#ay I .(O;12F )

a) _ac $Rnh a"nh c#"a cac $[nh ' .b) Tm a"nh c#"a ' G#a phep G#ay I

(O;12F )

b b b= = =

o

o

o+ia"i

a) > O' = O = O Ca3'O O O' 12F nen I :' '(O;12F )

b) I : ' '(O;12F )

1/ -@1P ho hnh C#ong 'B 9am O .a) Tm a"nh c#"a $ie%m G#a phep G#ay I .

(' ; F )b) Tm a"nh c#"a $D53ng 9haEng G#a phep G#ay I

(O ; F )

AB : a) +oHi = I () 9h '=' Ca(' ; F )

o

o

o b3' F nen '

C#ong can $[nh ' co$D53ng cao 'B . Bo $o: B a39r#ng $ie%m c#"a .b) Ta co: I () Ca3I () I () B.

(O ; F ) (O ; F ) (' ; F )

=

= = =o o o

o

= =

o

o o

1 ho hnh C#ong 'B 9am O . M a39r#ng $ie%m c#"a ' a39r#ng $ie%m c#"a O' . Tm a"nh c#"a 'M G#a phep G#ay I .

(O;F )AB : I (') B I (M) M a39r#ng $ie%m c#"a '

(O;F ) (O;F )`

= = o o

B .

I () a39r#ng $ie%m c#"a OB . Bo $o: I ( 'M) BM (O;F ) (O;F )

1L -1.1, P ho hnh #Hc giac $eS# 'B O a39am $D53ng 9ro3n ngoaHi 9iep c#"a no. Tm a"nh c#"a

O' G#a phep 453i hnh co$D5Hc baYng cach 9hDHc hie?n ien 9iep phep G#ay 9am O

= = =

= = =

###r

###ro

o o o###r ###r ###r

o

o

O

O (O;/F )

(O;/F ) (O;/F ) (O;/F )

O O O

goc /F Ca3phep9Rnh 9ien T .

AB :+oHi = T I . _e9 :

I (O) OI (') I () .

T (O) T () OT () B

>a?y : (O) = (') = O

`

`

` () = B ( O') = OB

- 2, -


26/39


o

1 ho hnh #Hc giac $eS# 'B 9heo chieS# 4D5ng O a39am $D53ng 9ro3n ngoaHi 9iep c#"a no. a9r#ng $ie%m c#"a ' .

a) Tm a"nh c#"a ' G#a phep G#ay I .(O ; 12F )

b) Tm a"

=

o

o

o

o

nh c#"a 'O G#a phep G#ay I .( ; /F )

AB :a) I bien ' aSn D5H9 9ha3nh B 9r#ng $ie%m

(O ; 12F ) 9ha3nh 9r#ng $ie%m ] c#"a B nen I ( ') ] .

(O ; 12F )b) I bien

( ; /F )

`

` 'O aSn D5H9 9ha3nh BO .

1, ho ba $ie%m ' 9heo 9hD9DH 9ren 9haEng ha3ng . >eXc#3ng mo?9 phWa 4DHng hai 9am giac $eS# ' Ca3 . +oHi M Ca3 9D5ng Dng a3hai 9r#ng $ie%m c#"a ' Ca3 . hDng minh raYng : M a39am giac $eS# .AB :_e9 phep G#ay I .Ta co: I (') I ()

(; /F ) (; /F ) (; /F )

I (') .(; /F )Bo M a39r#ng $ie%m c#"a ' a39r#ng $ie%m c#"a nen :

I (M) M(; /F )

= =

=

=

o o o

o

o= Ca3M /F M a39am giac $eS# .

= o

21 -1.1 P ho nD"a $D53ng 9ro3n 9am O $D53ng 6Wnh . 0ie%m ' chaHy 9ren nD"a $D53ng 9ro3n $o.BDHng CeSphWa ngoa3i c#"a ' hnh C#ong ' . hDng minh raYng : chaHy 9ren nD"a $

o

o

D53ng co$Rnh .AB : +oHi =I (') .


27/39


2* ho hai $D53ng 9ro3n (O) Ca3(O ) baYng nha# Ca3caJ9 nha# 5"' Ca3 .TD3$ie%m co$Rnh 6e"ca9 9#yen 4i $o?ng M C5i (O) M Ca3 caJ9(O ) 9aHi M Ca3 . hDng minh $D53ng 9haEng

M #on #on $i G#a mo?9$ie%m co$Rnh.+ia"i_e9 phep G#ay 9am ' goc G#ay ('O; 'O ) = bien (O) 9ha3nh (O ) .> MM Ca3 G#a nen ('O;'O ) = ('M;'M ) = (';' ) .

I#a phep G#ay I : M

('; )

M Ca34o $oI

M M 0D53ng 9haEng M G#a $ie%m co$Rnh nen $D53ng 9haEng M G#a$ie%m co$Rnh a3a"nh c#"a G#a I('; )

=

o

2, ho hai hnh C#ong 'B Ca3+a) Tm a"nh c#"a '+ 9rong phep G#ay I .

(; F )

b) +oHi M aSn D5H9 a39r#ng $ie%m c#"a '+ Ca3 .hDng minh M C#ong can .

+ia"i'

a) >(';

=

= =

=

o o

o o

o o

o

+ Ca3

) F (+;) F

I :' + I : '+ (; F ) (; F )

b) I :'+ I :M M Ca3(M;) = F(; F ) (; F )

M C#ong can 9aHi .

2/ ho ' . I#a $ie%m ' 4DHng hai 9am giac C#ong can ' Ca3' . +oHi M a39r#ng $ie%m c#"a Ca3gia"&D"'M = A . hDng minh : 'A a3$D53ng cao c#"a ' .

o

o

o

AB :_e9 phep G#ay I : ' = ' ' = 'B nen y ra : I bien aSn D5H9 9ha3nh B

(';F )0Nngh

nen goHi 9r#ng $ie%m < c#"a B 9h


28/39


2L -1.1LP ho ' . BDHng CeSphWa ngoa3i c#"a 9am giac cac hnh C#ong ]'M'Ca3goHi O@I aSn D5H9 a39am $oi xDng c#"a ch#ng .

a) +oHi B a39r#ng $ie%m c#"a ' . hD

o

ng minh raYng : BO@ C#ong can 9aHi B . b) hDng minh raYng : 'O @I Ca3'O = @I .AB :a) > : ' = I (M) M = ' Ca3M ' .

(;F )`

MaZ9 6hac : B@1

M BO2

'

B@ =Ca3 BO BO@ C#ong can 9aHi B .

=

o o(B;F ) (B;F )

b) TD3ca# a) y ra :I I

O @' I O' Ca3 @I.

2 ho ' cocac $[nh 6W hie?# 9heo hD5ng am . BDHngCeSphWa ngoa3i 9am giac $ocac hnh C#ong 'B Ca3eSphWa ngoa3i 9Dgiac 4DHng cac 9am giac $eS# 'M B@ . >eSphWa 9rong9D giac 4DHng hai 9am giac $eS# Ca3'B< . hDng minh : M@< a3hnh bnh ha3nh .

A

=

=

o

o

o

o

(;F )

(B;F )

/FB : _e9 phep G#ay I : M ' I

M ' M ' (1)

/F_e9 phep G#ay I : @ < 'BI

@< ' @< ' (2)TD3 (1) (2) y ra : M = @< .8W #a?n 9D5

ng 9DH : M< = @ M


29/39


=

o o(;/F ) (;/F )

!1 ho ' . >eSphWa ngoa3i 9am giac 4DHng ba 9am giac $eS#' ' ' . hDng minh raYng : '' $oSng G#y .1 1 1 1 1 1AB :

I I+[a &D"'' . _e9 : ' ' 1 1 1 1

' '1

b

b

= =

=

o o o

o

(;/F )I (' '; ) /F '] /F (1)1 1 1 1

8ay 9ren $ie%m &ao cho : = ' . > ' /F ' $eS# .1

o o o(';/F ) (';/F ) (';/F )I I I

_e9 : 1 1 > : 9haEng ha3ng nen 9haEng ha3ng1 1 '' $oSng G#y .1 1 1

!2 hDng minh raYng cac $oaHn 9haEng noi 9am cac hnh C#ong 4DHng9ren cac caHnh c#"a mo?9 hnh bnh ha3nh CeSphWa ngoa3i h5Hp 9ha3nhmo?9 hnh C#ong .AB : +oHi a39am c#"a1 2 ! *

=

= = =

= o

o

o

(;F )

hnh C#ong caHnh 'BB' .

B#3ng phep G#ay I(;F ): . > ' B1 !

Ca3B ' *, . Ma3B NN ' ! 1 ! 1 ! 1I

>a?y : Ca3 .! 1 2 1 2 ! 2 1 2 !8y#a?n 9D5ng 9

DH 9a co: a3mo?9 hnh C#ong .1 2 ! *

Vn 6 HJI H!NH OdNG NHJUJ. KIN THL ON

1 08 : e# ' Ca3' a3hai 9am giac baYng nha# 9h co phep 453i hnh bien ' 9ha3nh ' .

2 TWnh cha9 :1. e# 9hDHc hie?n ien 9iep hai phep 453i hnh 9h $D5Hc mo?9 phep 453i hnh .2. Aai hnh goHi a3baYng nha# ne# cophep 453i hnh bien hnh na3y 9ha3nh hnh 6ia .

O. O"I TP

1 ho hnh chDXnha?9 'B . +oHi A 9heo 9hD9DH a39r#ng $ie%m c#"a cac caHnh 'B. AaXy 9m mo?9 phep 453i hnh bien ' 9ha3nh A .AB :

ThDHc hie?n ien 9iep phep 9Rnh 9ie

=

###r

###r ###r

n 9heo ' Ca3phep $oi xDng G#a $D53ng 9haEng A

T :' A T ( ') A' '

`

=

=

= =

###r

###ro

0 : A A 0 ( A) AA A 0 :T ( ') AA ' Bo $o: 0 T ( ') A ' AA '

`

`

- 2 -


30/39


2 ho hnh chDXnha?9 'B . +oHi O a39am $oi xDng c#"a no; +A] 9heo 9hD9DH a39r#ng $ie%m c#"acac caHnh 'BB''AO+ . hDng minh raYng : Aai hnh 9hang ']O Ca3+] baYng

###r

nha# .AB :

@hep 9Rnh 9ien 9heo 'O bien 'O aSn D5H9 9ha3nh O] . @hep $oixDng G#a 9r#Hc c#"a O+ bien O] aSn D5H9 9ha3nh +].

TD3$oy ra phep 453i hnh co$D5Hc baYng cach 9hDHc hie?n ien 9iep haiphep bien hnh 9ren &eXbien hnh 9hang ']O 9ha3nh hnh 9hang +] . Bo $ohai hnh 9hang ay baYng nha# .

o

r! -1.2FP Trong mpOxy cho # = (!;1) Ca3$D53ng 9haEng (4) : 2x y = F . Tm a"nh c#"a (4) G#a phep453i hnh co$D5Hc baYng cach 9hDHc hie?n ien 9iep phep G#ay I Ca3phep 9Rnh 9ien

(O;F )rT .#

= =

+ + =

ro

o o

(O;F ) #I T

AB : @@ : 4 4 4 +oHi 4 I (4) . > 9am O 4 nenI (O) O 4 .

(O;F ) (O;F ) MaZ9 6hac : 4 4 4 :x 2y F ( F) ma34 G#a O nen = F 4: x 2y = F

ach 6hac : hoHn

`

= + = = = =

= + = + = +

=

o

o o o o o

o o o o o

(O;F )I

M(1;2) 4 M 4 .

x OM co&( F ) x OM co& co&F OM&in &inF x xco&F y&inF Ta co: M

y OM &in( F ) y OM &in co&F OM co& &inF y yco&F x&inF

= = = = = + = + + =

= + = = = + = + + =

o o

o o

r####r

rr

x 1co&F 2&inF x 2 M ( 2;1)y 1y 2co&F 1&inF

+oHi 4 T (4) 4 NN 4 4 : x 2y F .#

x x ! x ! +oHi O T (O) OO = # O(!;1) .# y y 1 y 1 > 4 O ! 2 F

`

= + == + =or o , 4 :x 2y , F

>a?y :T I (4) (4):x 2y , F# (O;F )

+ + =

r2 2* Tm a"nh c#"a $D53ng 9ro3n () : x y 2x *y * F co$D5Hc baYng cach 9hDHc hie?n ien 9iep phep

9Rnh 9ien 9heo # = (!; 1) Ca3phep 0 .Oy

+ + =2 2 0Q : ( ) : (x *) (y !)

+ + =

o

2 2, Tm a"nh c#"a $D53ng 9ro3n () : x y /x 2y / F co$D5Hc baYng cach 9hDHc hie?n ien 9iep phepG#ay I Ca3phep 0 .Ox(O;F )

AB : () co9am (!;1) b6 : K = 2 .


31/39


o

(O;

I Ca3phep $oi xDng 0 . Tm 9oaH $o?cac $[nh c#"a ' .Ox 1 1 1(O; F )AB :a) +oHi M aSn D5H9 a3hnh chie# c#"a ' 9ren OxOy 9h

M( !;F)(F;2).I


32/39


Vn 7 PHP V Te

=###r ###r

1 0 : ho $ie%m co$inh Ca3mo?9 &o6 F . @hep CR9DH 9am 9[&o6 .6 a3phep bien hnh bien moi $ie%m M 9ha3nh $ie%m M &ao cho M 6 M.

= =

6

62 ie%# 9hDc 9oHa $o?: ho (x ;y ) Ca3phep CR9DH > .o o x= 6x (1 6)x> 6 o M(x;y) M > (M) (x ;y ) 9h y= 6y (1 6)yo

= =#####r ####r

! TWnh cha9 :6 61. M > (M) > () 9h M = 6M M = \6\.M

2. ien ba $ie%m 9haEng ha3ng 9ha3nh ba $ie%m 9haEng ha3ng Ca3ba"o 9oa3n 9hD9DH c#"a cac $ie%m 9D5ng Dng .!. ien mo?9 $D53ng 9haEng 9ha3nh mo?9 $D53ng 9haEng &ong &ong hoaZc 9r#3ng C5i $D53ng 9haEng $aXcho .

*. ien mo?9 9ia 9ha3nh 9ia .,. ien $oaHn 9haEng 9ha3nh $oaHn 9haEng ma3$o?4a3i $D5Hc nhan en \6\ ./. ien9am giac 9ha3nh 9am giac $oSng 4aHng C5i no.. 0D53ng 9ro3n coban 6Wnh K 9ha 3nh $D53ng 9ro3n coban 6Wnh K =\6\.K .L. ien goc 9ha3nh goc baYng no.

O . O"I TP

1 Tm a"nh c#"a cac $ie%m &a# G#a phep CR9DH 9am 9[&o6 F : a) '(1;2) (!; 1) 6 = 2 .

= ' ( 1;,)

b) (2; !) ( 1; 2)6 ! . ( 1F;1)1

c) (L;!) (2;1) 6 = .2

(,;2)

2 1 1 4) @( !;2)I(1;1)K(2; *) O6 = 1N! @(1; )I( ; )! ! !

= = + = + = = =

###r ##r(;2)

2 *K ( ; )! !

> x ! *AB : a) +oHi : '(1;2) ' (x;y ) ' 2' (x !;y 1) 2( 2;!)y 1 /

x 1 ' ( 1;,) .y ,

2 ho ba $ie%m '(F;!)(2; 1)( 1;,) . ToSn 9aHi hay 6hong 9oSn 9aHi mo?9 phep CR9DH 9am ' 9[&o6 bien 9ha3nh 7AB : +[a &D"9oSn 9aHi mo?9 phep CR9DH 9am ' 9[&o6 bien 9ha3

= = = =

###r ###r(';6)

nh .> 11 6(2) a?y : ToSn 9aHi phep CR9DH > : .1('; )

2! ho ba $ie%m '( 1;2)(!;1)(*;!) . ToSn 9aHi hay 6hong 9oSn 9a

=

###r ###r(';6)

Hi mo?9 phep CR9DH 9am ' 9[&o6 bien 9ha3nh 7AB : +[a &D"9oSn 9aHi mo?9 phep CR9DH 9am ' 9[&o6 bien 9ha3nh .

>


33/39


* ho OM . BDHng a"nh c#"a M G#a phep CR9DH 9am O 9[&o6 9rong moi 9rD53ng h5Hp &a# :1 !

a) 6 = ! b) 6 = c) 6 =2 *

+ia"i!a) @hep CR9DH > :M M O = =

=

####r ####r ####r ###r

###r ####r ###r

9h 9a coOM !OMO !O

1N2b) @hep CR9DH > :M A < 9h A< a3$D53ng 9r#ng bnh c#"a OM .O!!N*c) @hep CR9DH > :M @ I 9h 9a coO@ OMOIO *

= ###r!O

*

, ho hnh bnh ha3nh 'B (9heo chieS# 6im $oSng hoS) co9am O . BDHng :a) 'Vnh c#"a hnh bnh ha3nh 'B G#a phep CR9DH 9am O 9[&o6 = 2 .

b) 'Vnh c#"a hnh bnh ha3nh 'B G#a phep CR9

= =

=

####r ###r

####r ###r

####r ###r

1DH 9am O 9[&o6 = .

2+ia"i

2

a) +oHi > :' ' 9h O' 2O'O 9h O 2O

9h O 2O

B B 9h O

=

=

=

####r ###r

l l

###r ###r

###r ###r

B 2O2 > : 'BM ' B .O

Ta CeX: 'NN' NN BNN B B' NNB '11N2b) +oHi > :' @ 9h O@ O'O 2

1 I 9h OI O2

=

=

###r ###r

###r ###r

l l

1 K 9h OK O

21

B Q 9h OQ OB2

1N2 > : 'BM @IKQ .O Ta CeX: 'NN@INNIKBNNKQB' NNQ@ .

a / ho ' co' = * ' = / 'B a3phan giac 9rong c#"a ' c#"a ' (B ) . >5i gia9rRna3o

c#"a 6 9h phep CR9DH 9am B 9[&o6 bien 9ha3nh .AB :

Theo 9Wnh cha9 c#"a phan gi a

= = = =

###r###r ###r

###r###r ###r

(B; !N2)

ac 9rong c#"a ' 9a co:>B ' * 2 !

B B .' / ! 2B

Bo B Ca3B ngD5Hc hD5ng .

- !! -


34/39


=

ho ' C#ong 5"' Ca3' = / ' = L . @hep CR9DH > bien 9ha3nh 9ha3nh .!('; )

2 ') $#ng C .

! ! 2 2 ) &ai C : = ' ' 1,2 2

`

`

= = =

=

1 ! !.'.' .'. .'Q ' 2 2 2 ) $#ng C : .

1Q '.' *' .'.'2

h# Ci ' !B) $#ng C :

h# Ci ' 2

`

`

L ho ' cohai $[nh a3 Ca3 co$Rnh co3n $[nh ' 4i $o?ng 9ren $D53ng 9ro3n (O) cho 9rD5c .Tm 9a?p h5Hp cac 9roHng 9am c#"a ' .

=##r ##r1

AB : +oHi a39r#ng $ie%m c#"a . Ta co co$Rnh . e# + a39roHng 9am c#"a ' 9h + ' .!

1N! >a?y + a3a"nh c#"a ' G#a phep CR9DH > .Ta?p h5Hp $ie%m ' a3$D53ng 9ro3n (O) nen 9a?p h5Hp + a3$D53ng 9ro3n (O) $ochWnh a3a"nh c#"a $D53ng 9ro3n

1N!(O) G#a phep CR9DH > . Trong mpOxy cho $ie%m '( 1;2) Ca3$D53ng 9haEng 4

$i G#a ' cohe?&ogoc baYng 1 . +oHi a3$D53ng9haEng 4i $o?ng 9ren 4 . +oHi a3$ie%m &ao cho 9Dgiac O' a3hnh bnh ha3nh .Tm phD5ng 9rnh 9a?ph5Hp :a) ac 9am $oi xDng c#"a hnh bnh ha3nh .b) ac 9roHng 9am + cac 9am giac ' .

= + = +

+

gg

AB :a)

I#a '( 1;2) ('): ('):y 2 1(x 1) y x !

A&g : 6 =1

1 >a?y chaHy 9ren 4 9h chaHy 9ren 4 NN 4 Ca3$i G#a 9r#ng $ie%m M( ;1) c#"a $oaHn O' .2

! >a?y 4 : x y =F .

2

b) Ta

`

`

` = = ==

+

###r ###r2 2 *2N! 2N!co: O+ O + > () . >a?y + chaHy 9ren $9 4 NN 4 Ca3G#a $ie%m ( ; ) > (').O O! ! ! 4 : x y 2 =F .

- !* -


35/39


+

1F Tm a"nh c#"a cac $D53ng 9haEng 4 G#a phep CR9DH 9am 9[&o6 :2

a) 4 : !x y , =F >(O; ) 4 :x !y 1!

=

+ + =+

F F

b) 4 : 2x y * =F >(O;!) 4 :2x y 12 Fc) 4 : 2x y * =F >(; 2) C5i ( 1;2) + + =

+ + = 4 :2x y L F

4) 4 : x 2y * =F >(;2) C5i (2; 1) 4 :x 2y L F

11 Tm a"nh c#"a cac $D53ng 9ro3n () G#a phep CR9DH 9am 9[&o6 : (o2 cach gia"i )2 2a) () : (x 1) (y 2) =, >(O; 2) () : (x 2) + + + 2 2(y *) = 2F2 2 2 2b) () : (x 1) (y 1) =* >(O; 2) () : (x 2) (y 2) = 1/2 2c) () : (x !) (y 1) =, >(; 2) C5i (1;2)

+

+ +

+ + 2 2 () : (x !) (y L) =2F

12 Tm phep CR9DH bien 4 9ha3nh 4 :x y

a) 4 : 14 :2x y / F>(O;6)2 *

+ +

= =2

6 = .!

AB : 4 : 2x y * F NN 4 : 2x y / F . 8ay '(2;F) 4(!;F) 4 .

! > : phep CR9DH >(O;6) : ' O 6O' . > : O'=(2;F)O (!;F) O O'2

= =

= = =###r ###r ###r ###r ###r ###r

! !>(O; ) >(O; )

2 2 >a?y : ' 4 48D# y: > O' 9haEng ha3ng nen 9a choHn ch#ng c#3ng naYm 9ren mo?9 $D53ng 9haEng . 0e%$5n gia"n 9a choHnch#ng c#3ng naYm 9ren Ox hoaZc Oy

.

+ + = + =

= =

2 2 2 2b) ( ):(x *) y 2 ; ( ):(x 2) (y !) L >(; 2)( 2;1)1 2 AB :

( ) co9am ( *;F)K 2 ( ) co9am (2;!)K 2 21 1 1 2 2 2>(;6)

+[a &D":( )1 `` ( ) 9h :2

= = = =

= =

=

g###r ##r

g

K2 K \6 \K \6 \ 2 6 22 1 K1 6 9h2 1 6 = 2 . +oHi (x ;y ) 9h (2 x ;! y ) 2( * x ; y ) ( 2;1)o o o o o o 6 = 2 . +oHi (x ;y ) 9h (2 x ;! y ) 2( * x ; y ) ( 1F; !)o o o o o o

>a?y co2 phep CR9DH bien ( ) ( ) a3>(; 2) C5i ( 2;1) hoaZc >(;2) C5i ( 1F; !)1 2

+ + 2 2 2 21! Trong mpOxy cho 2 $D53ng 9ro3n ( ): (x 1) (y !) = 1 Ca3( ) : (x *) (y !) = * .1 2a) _ac $Rnh 9oaH $o?9am CR9DH ngoa3i c#"a hai $D53ng 9ro3n $o.b) >ie9 phD5ng 9rnh cac 9iep 9#yen c

= =

=###r ##r

h#ng ngoa3i c#"a hai $D53ng 9ro3n $o.AB : ( ) co9am (1;!) b6 : K 1 ; ( ) co9am (*;!) b6 : K 2 .1 1 1 2 2 2

a) +oHi a39am CR9DH ngoa3i c#"a ( ) Ca3( ) 9a co: 6 C5i1 2 2 1 = = K 226 = 2 ( 2;!)K 11

+ + =

= =

b) Tiep 9#yen ch#ng ngoa3i c#"a hai $D53ng 9ro3n a39iep 9#yen 9D3 $en ( ).1 +oHi $9 $i G#a Ca3cohe?&ogoc 6 :y ! = 6(x2) 6y y ! 26 F .

1 9iep x#c ( ) 4( ; ) K 61 1 1

2 2

+ + = + + =

: 2.x *y 12 ! 2 F1: 2.x *y 12 ! 2 F2

- !, -


36/39


n n

=1* ho $D53ng 9ro3n (OK) $D53ng 6Wnh ' . Mo?9 $D53ng 9ro3n (O ) 9iep x#c C5i (OK) Ca3$oaHn ' 9aHi

B $D53ng 9haEng B caJ9 (OK) 9aHi . hDng minh raYng : ' .AB : a39am C`

R9DH c#"a 2 $D53ng 9ro3n (O) Ca3(O) .B (O ) (O) Ca3ba $ie%m B 9haEng ha3ng .

+oHi K a3ban 6Wnh c#"a $D53ng 9ro3n (O ) 6hi $o:

KK > :O O B

O NN OB O ' (>

`

j n n

=

OB ')

a39r#ng $ie%m c#"a ' ' .

1, ho hai $D53ng 9ro3n (OK) Ca3(O K ) 9iep x#c 9rong 9aHi ' (K d K ) .0D53ng 6Wnh G#a ' caJ9 (OK) 9aHi Ca3caJ9 (O K ) 9aHi . Mo?9 $D53ng9haEng 4i $o?ng G#a ' caJ9 (O K) 9aHi M Ca3ca

=:

J9 (O K ) 9aHi . Tm G#yX9Wch c#"a = M .AB :

Ta co: M NN . Aai M . Bo $o:M M

=

= = = =

+ +

= + = = + +

:

##r ####r

' Aai ' 'M . Bo $o:

' M ' 2K K K

M ' 2K K M K KK

>(;6 ) K K K K M M : M K K K K

>a?y : Ta?p h5Hp cac $ie%m a3$D53ng 9ro3n ( ) CR9DH c#"a $D53ng

= +K 9ro3n (OK) 9rong phep CR9DH >(;6 ) .K K

1/ ho ' . +oHi ] . M 9heo 9hD9DH a39r#ng $ie%m c#"a ' ' Ca3] . 0D53ng 9ro3n ngoaHi 9iep 9am Oc#"a '] caJ9 'O 9aHi ' . +oHi M a3chan $D53ng C#ong goc haH 9D3' x#ong

. hDng minh raYng : '

M M 9haEng ha3ng .

= =

###r ##r ###r ##rAB :+oHi M a39r#ng $ie%m .Ta co: ' 2' Ca3' 2']1

>(';2)TD3 $o: '] ' . :O ' M M OM ] ' M .(';2) 1 1 hD 9he: M M 'MM 9haEng ha3ng ( C 'M1

M 9haEng ha3ng )1

1 ho ' . +oHi ' 9D5ng Dng a39r#ng $ie%m c#"a '1 1 1'.


37/39


1L ho hai $D53ng 9ro3n (O K ) Ca3(O K ) ngoa3i nha#1 1 2 2K K . Mo?9 $D53ng 9ro3n (O) 9hay $o%i 9iep x#c ngoa3i1 2C5i (O ) 9aHi ' Ca39iep x#c ngoa3i C5i (O ) 9aHi . hDng1 2minh raYng : 0D53ng 9haEng ' #on #on $i G#a mo?9 $ie%mco$Rnh .

####r ###r####r ###r

###r

AB:' a39am CR9DH bien (O ) 9ha3nh (O) : 'O Ca3'O ngD5Hc hD5ng .1 1 a39am CR9DH bien (O) 9ha3nh (O ) : 'O Ca3'O ngD5Hc hD5ng .2 1a?y : 'O Ca3O ngD5Hc hD5ng . hD Ca?y ' hay c#Xng a31 2' pha"i $i G#a 9am a39am CR9DH ngoa3i c#"a (O ) Ca3(O ) .1 2

1 ho ' . gD53i 9a m#on $Rnh ba $ie%m ' aSn D5H9 9ren cac caHnh '' &ao cho ' $eS# Ca3' ' ' Ca3 ' .1. +oHi < aSn D5H9 a3chan cac $D53ng cao

=

####r ###r

pha9 x#a9 9D3' .

2N! 2N! 2N! 0aZ9 : => (')' => ()=> (). 22N! a) ghie?m aHi raYng : ' => () Ca3 < . !

b) Q#y ra raYng : ' $eS# .2. hDng minh raYng 9rDHc

9am A c#"a ' c#Xng a39roHng 9am c#"a ' .AB :

a !Trong ' $eS# cac $D5ng cao : ' = = < = .(a a3caHnh c#"a ')

2Ca3< aSn D5H9 a39r#ng $ie%m cac caHnh .

1. a) > ' => = + = =

= + = = =

####r ###r ###r ####r ###r ####r ###r

###r ###r ###r ####r ###r ####r ###r ###r

2 2 1 22N! 2N!() ' ' ( ) ' . >a?y : ' => () . ! ! 2 !2 2 1 22N! 2N! > = > (') ' ' ' ' ' ' '< = > (). '! ! ! !

=####r ###r2N! 2N!

' '> > 2 >a?y : < < .!

=

g####r ###r

g

NN< c#3ng '2

b) Ta co:


38/39


Vn 8 PHP #NG D$NGJ. KIN THL ON

1 0 : @hep bien hnh goHi a3phep $oSng 4aHng 9[&o6 (6 dF) ne# C5i hai $ie%m ba9 6 M Ca3a"nh M a3a"nh c#"a ch#ng 9a coM =6.M .

2 08 : MoHi phep $oSng 4aHng 9[&o6 (6dF) $eS# a3h5Hp 9ha3nh c#"a mo?9 phep CR9DH 9[&o6 Ca3mo?9 phep

453i hnh B.! Ae?G#a": (TWnh cha9 ) @hep $oSng 4aHng :1. ien ! $ie%m 9haEng ha3ng 9ha3nh ! $ie%m 9haEng ha3ng (Ca3ba"o 9oa3n 9hD9DH ) .2. ien $D53ng 9haEng 9ha3nh $D53ng 9haEng .!. ien 9ia 9ha3nh 9ia .*. ien $oaHn 9haEng 9ha3nh $oaHn 9haEng ma3$o?4a3i $D5Hc nhan en 6 ( 6 a39[&o$oSn

g 4aHng ) .,. ien9am giac 9ha3nh 9am giac $oSng 4aHng C5i no( 9[&o6)./. ien $D53ng 9ro3n coban 6Wnh K 9ha3nh $D53ng 9ro3n coban 6Wnh K =6.K .. ien goc 9ha3nh goc baYng no.

*

Aai hnh $oSng 4aHng :0 : Aai hnh goHi a3$oSng 4aHng C5i nha# ne# cophep $oSng bien hnh na3y 9ha3nh hnh 6ia .

A $oSng 4aHng + $oSng 4aHng : A +

O.O"I TP

1 ho $ie%m Ma) BDHng a"nh c#"a phep $oSng 4aHng a3h5Hp 9ha3nh c#"a phep $oi xDng 9r#Hc 0 Ca3phep CR9DH > 9am O a C5i O a 9[&o6 =2 .b) BDHng a"nh c#"a phep $oSng 4aHng a3

o

2a O

h5Hp 9ha3nh c#"a phep CR9DH > 9am O 9[&o6 = ! Ca3phep G#ay

9am C5i goc G#ay =F .

+ia"i0 >

a) +oHi : M M M1 2 M (a) 9h M M Ca3M a39r#ng $ie%m OM1 2 M (a) C

``

gg

gg

a3O M 9h :1 a a39r#ng 9rDHc $oaHn MM1 M a39r#ng $ie%m $oaHn OM1 2 M (a) Ca3O M 9h :1 a a39r#ng 9rDHc $oaHn MM1 M a39r#ng $ie%m $oaHn OM1 2

b) +oH

`

= =

o

#####r ####r o

! FO > Ii M M M .


39/39


2 ho ' co$D53ng cao 'A . A 5"9ren $oaHn . ie9 'A =* A =2 A =L . @hep $oSng 4aHng bien A' 9ha3nh A' . $D5Hc h5Hp 9ha3nh b5"i hai phep bien hnh na3o 4D5i $ay 7

') @###r

1hep $oi xDng 9am A Ca3phep CR9DH 9am A 9[&o6 = .

2) @hep 9Rnh 9ien 9heo ' Ca3phep CR9DH 9am A 9[&o6 =2 .) @hep CR9DH 9am A 9[&o6 =2 Ca3phep G#ay 9am A goc (A

;A') .B) @hep CR9DH 9am A 9[&o6 =2 Ca3phep $oi xDng 9r#Hc .AB :

2 @hep > Ca3I(A; ) C5i =(A;A') : ' ' A >a?y : a3phep $oSng 4aHng h5Hp 9ha3nh b5"i > Ca3I bien A

' 9ha3nh A' .

+ =

! ho hnh bnh ha3nh 'B co9am O . Tren caHnh ' ay $ie%m &ao cho ' 2 F Ca3goHi + a39roHng 9am c#"a 'B . a3phep $oSng 4aHng bien '+ 9ha3nh OB . $D5Hc h5Hp 9ha3nh

###r

b5"i hai phepbien hnh na3o &a# $ay 7') @hep 9Rnh 9ien 9heo +O Ca3phep CR9DH >(; 1) .

1) @hep $oi xDng 9am + Ca3phep CR9DH >(; ).2!

) @hep CR9DH >('; ) Ca3phep $oi xDng2

9am O .

2B) @hep CR9DH >('; ) Ca3phep $oi xDng 9am + .

!

=

=

###r ###rg

###r ##rg

g

g2N!

O'

AB :!

> + a39roHng 9am 'B nen 'O '+2

! Theo gia"9hie9 9a co: ' '] .

2 @hep $oi xDng 9am O bien ' 9ha3nh Ca3 9ha3nh B ( O a3ba9 bien )

0> ' ' g g

2N! 2N!O O' '0 0> > . + O O . B .

O!

>('; ) 02'+ 'O OB @hep $oSng 4aHng

. . . .. . . . . . HT . . . . . . ..

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Chuyen de Phep Bien Hinh Trong Mp (1)