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ThS inh Xun Nhn

PHP TNH TINBi 1. Trong mt phng Oxy, cho vect ( )1; 2u = .

a) Tm nh ca ( )2;1A qua php tnh tin uTr .b) Tm nh ca ng thng 1 : 3 4 0d x y + = qua php tnh tin uTr .c) Tm ng thng l nh ca ng thng : 2 3 0d x y + = qua php tnh tin uTr .d) Tm nh ca ng trn ( ) ( ) ( )2 2: 2 3 4C x y+ + = qua php tnh tin uTr .

Giia) Gi ( ) 'uT A Ar , ( )' '; ' A x y .

Biu thc to ca php tnh tin ( )' ' 2 1 3 ' 3; 1' ' 1 2 1

x x a xA

y y b y

= + = + = = + = = .b) Cch 1.

Chn mt im ( ) 11;1 : 3 4 0M d x y + = .ng thng d1 i qua ( )1;1M v c vecto php tuyn ( )1; 3n = .Gi ( ) 'uT M Mr , ( )' '; 'M x y .Biu thc to ca php tnh tin uTr : ( )' ' 1 1 0 ' 0; 1' ' 1 2 1 x x a x M y y b y= + = + = = + = = .Gi ( ) '1 1uT d dr .ng thng '1d i qua ( )' 0; 1M v c vecto php tuyn ( )1; 3n = .

( ) ( )' '1 1:1 0 3 1 0 : 3 3 0d x y d x y + = = .Cch 2.Gi ( ) '1 1uT d dr .Gi ( ) ( ) '1 1; , ' '; 'M x y d M x y d sao cho ( ) 'uT M Mr = .Biu thc to ca php tnh tin uTr :

' ' ' 1

' ' ' 2

x x a x x a x x

y y b y y b y y

= + = = = + = = + .

( ) ( ) ( )1' 1; ' 2 : 3 4 0 ' 1 3 ' 2 4 0 ' 3 ' 3 0M x y d x y x y x y+ + = + + = =Vy '1 : 3 3 0d x y = .

c) Cch 1.Chn mt im ( )1; 1 : 2 3 0M d x y + + = .ng thng d i qua ( )1; 1M v c vecto php tuyn ( )2;1n = .Gi ( ) 'uT M Mr .Biu thc to ca php tnh tin uTr : ( )' ' 1 1 0 ' 0; 3' ' 1 2 3

x x a xM

y y b y

= + = + = = + = = .Theo bi ( )uT d = r .ng thng i qua ( )' 0; 3M v c vecto php tuyn ( )2;1n = .

( ) ( ): 2 0 3 0 : 2 3 0 x y x y + + = + + = .Cch 2.Theo bi ( )uT d = r .Gi ( ) ( ); , ' '; 'M x y d M x y sao cho ( ) 'uT M Mr = .

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ThS inh Xun Nhn

Biu thc to ca php tnh tin uTr :' ' ' 1

' ' ' 2

x x a x x a x x

y y b y y b y y

= + = = = + = = + .( ) ( ) ( )' 1; ' 2 : 2 3 0 2 ' 1 ' 2 3 0 2 ' ' 3 0M x y d x y x y x y+ + + = + + + = + + =

Thay ' 1x x v ' 2y y + vo phng trnh d, ta c( ) ( )2 3 0 2 ' 1 ' 2 3 0 2 ' ' 3 0 x y x y x y+ = + + + = + + = .

Vy : 2 3 0x y+ + = .d) ng trn ( ) ( ) ( )2 2: 2 3 4C x y+ + = c tm ( )2; 3I v bn knh 2R = .

Gi ( ) ( )', ' '; 'uT I I I x yr = .Biu thc to ca php tnh tin uTr : ( )' ' 2 1 3 ' 3; 5' ' 3 2 5

x x a xI

y y b y

= + = + = = + = = .Gi ( ) ) ( )uT C Cr .ng trn ( )'C c tm ( )' 3; 5I v bn knh 2R = .

( ) ( ) ( )2 2' : 3 5 4C x y + + = .Bi 2. Trong mt phng Oxy, cho vect ( )2;4u = .

a) Tm to im M sao cho ( )3;1A l nh ca M qua php tnh tin uTr .b) Tm ng thng 1 c nh qua php tnh tin uTr l 1 : 2 3 5 0d x y + = .c) Tm ng trn ( )1C c nh qua php tnh tin uTr l ( ) ( ) ( )2 2: 1 2 4C x y+ + = .

Giia) Theo bi ta c ( )uT M Ar .

Biu thc to ca php tnh tin uTr :

( )' ' 3 2 1 1; 3' ' ' 1 4 3

x x a x x aM

y y b y y b

= + = = + = = + = = = .b) Cch 1.

Chn mt im ( ) 11;1 : 2 3 5 0M d x y + = .ng thng d1 i qua ( )1;1M v c vecto php tuyn ( )2; 3n = .Gi ( );N x y l im sao cho ( )uT N Mr .Biu thc to ca php tnh tin uTr :

( )' ' 1 2 1 1; 3' ' ' 1 4 3

x x a x x aN

y y b y y b

= + = = + = = + = = = .Theo bi ( )1 1uT d=r .ng thng 1 i qua ( )1; 3N v c vecto php tuyn ( )2; 3n = .

( ) ( )1 1: 2 1 3 3 0 : 2 3 11 0 x y x y + = = .Cch 2.Theo bi ( )1 1uT d=r .Gi ( ) ( )1 1; , ' '; 'M x y M x y d sao cho ( ) 'uT M Mr = .Biu thc to ca php tnh tin uTr :

' ' 2

' ' 4

x x a x x

y y b y y

= + = = + = + .( ) ( ) ( )1' 2; 4 : 2 3 5 0 2 2 3 4 5 0 2 3 11 0M x y d x y x y x y+ + = + + = = .

Vy 1 : 2 3 11 0x y = .

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ThS inh Xun Nhn

Gii.Cu a v b dnh cho c gi.

c) Theo bi ( ) 'uT d dr .Gi ( ) ( ); , ' '; ' 'M x y d M x y d sao cho ( ) 'uT M Mr = .Biu thc to ca php tnh tin uT

r

:

' ' ' 2

' ' ' ' 3

x x a x x a x x

y y b y y b y y

= + = = = + = = + .( ) ( ) ( )' 2; ' 3 : 2 4 0 2 ' 2 ' 3 4 0 2 ' ' 3 0M x y d x y x y x y+ + = + + = = .' : 2 3 0d x y = .

Tip theo ta c ( )' "vT d dr = .Gi ( ) ( ); ', ' '; ' "N x y d N x y d sao cho ( ) 'vT N Nr = .Biu thc to ca php tnh tin vTr :

' ' ' 1

' ' ' 3

x x a x x a x x

y y b y y b y y

= + = = = + = = .( ) ( ) ( )' 1; ' 3 ' : 2 3 0 2 ' 1 ' 3 3 0 2 ' ' 2 0 N x y d x y x y x y = = = .

Vy " : 2 2 0d x y = .d) Gi ( ) ) ( )'uT C Cr .

ng trn ( ) ( ) ( )2 2: 2 2 9C x y+ + = c tm ( )2; 2I v bn knh 3R = .Gi ( )' '; ' I x y l nh ca ( )2; 2I qua php tnh tin uTr : ( ) 'uT I Ir .Biu thc to ca php tnh tin uTr : ( )' 2 2 4 ' 4; 5' 2 3 5

x x aI

y y b

= + = + = = + = = .Gi ( )" "; " I x y l nh ca ( )' 4; 5I qua php tnh tin vTr : ( )' "vT I Ir .Biu thc to ca php tnh tin uTr : ( )" 4 1 5 " 5; 2

" 5 3 2

x x aI

y y b

= + = + = = + = + = .ng trn (C) c tm ( )" 5; 2I v bn knh 3R = .

( ) ( ) ( )2 2

" : 5 2 9C x y + + = .Bi 5. Cho hnh bnh hnh ABCD c 2 nh A,B c nh, tm I thay i trn ng trn (O).Tm tp hp trung im ca cnh BC.GiiDo I l tm hnh bnh hnh ABCD, v M ltrung im BC nn ta c

1

2 IM ABuuur uuur= .

Suy ra M l nh ca I qua php tnh tin theo

vecto1

2

v ABr uuur= . M im I di ng trn ng

trn (O). Vy tp hp im M nm trn ngtrn (O) l nh ca (O) qua php tnh tin theo

vecto1

2v ABr uuur= .

Bi 6. Cho im A c nh trn ng trn (O,R) v im B lu ng trn ng trnny. Dng hai hnh bnh hnh OAMB, OABN. Hy tm tp hp 2 im M, N.Gii.

O'

M

D

C

O

B

A

I

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