Cac Dang Toan Ve Phep Bien Hinh

7/31/2019 Cac Dang Toan Ve Phep Bien Hinh
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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNH
PHP BIN HNH V PHP DI HNHPHP TNH TIN
I.Tm tt l thuyt :1. nh ngha : Trong mt phng , cho vc t ( );v a b . Php tnh tin theo vc t
( );v a b l php bin hnh , bin mt im M thnh mt im M sao cho 'MM v=K hiu : vTr .2.Cc tnh cht ca php tnh tin :a/ Tnh cht 1:
*nh l 1: Nu php tnh tin bin hai im M,N thnh hai im M,N thMN=MN.
b/ Tnh cht 2:* nh l 2: Php tnh tin bin ba im thng hng thnh ba im thng hng vkhng lm thay i th t ca ba im .
H QU :Php tnh tin bin ng thng thnh ng thng , bin mt tia thnh mt tia , binmt on thng thnh mt on thng bng n , bin mt tam gic thnh mt tam gic
bng n , bin mt ng trn thnh mt ng trn c cng bn knh , bin mt gcthnh mt gc bng n .3. Biu thc ta ca php tnh tin- Gi s cho ( );v a b v mt im M(x;y) . Php tnh tin theo vc t v bin im M
thnh im M th M c ta l :'
'
x a x
y y b
= + = +
4. ng dng ca php tnh tin
BI TON 1: TM QU TCH CA MT IMBi ton : Cho mt hnh H , trn hnh H c mt im M . Tm qu tch ca im Mkhi trn hnh H c mt im A thay i . ( Thng im A chy trn mt ng (C )cho sn ).Cch gii :- Da vo cc tnh cht bit , ta tm ra mt vc t c dnh nm trn hnh H ( Viiu kin : vc t ny c phng song song vi ng thng k qua A ).
- Sau da vo nh ngha v php tnh tin ta suy ra M l nh ca A qua php tnhtin theo vc t c nh .- Da vo tnh cht thay i ca A ta suy ra gii hn qu tch .
V d 1: Cho hai im B,C c nh nm trn (O,R) v mt im A thay i trnng trn . Chng minh rng trc tm ca tam gic ABC nm trn mt ngtrn c nh .
Gii
- K ng knh BB .Nu H l trc tm ca tam gic ABC th AH=BC. Do C,B cnh , cho nn BC l mt vc t c nh 'AH B C = . Theo nh ngha v php tnh
Bin son : Nguyn nh S - T: 02403833608 Trang 1

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHtin im A bin thnh im H . Nhng A li chy trn (O;R) cho nn H chy trnng trn (O;R) l nh ca (O;R) qua php tnh tin dc theo 'v B C=- Cch xc nh ng trn (O;R) . T O k ng thng song song vi BC . Sau dng vc t : OO' 'B C= . Cui cng t O quay ng trn bn knh R t tm O tac ng trn cn tm .
V d 2. Cho hnh bnh hnh ABCD c hai nh A,B c nh , cn nh C chy trnmt ng trn (O;R). Tm qu tch nh D khi C thay i .Gii :
- Theo tnh cht hnh bnh hnh : BA=DC AB CD = . Nhng theo gi thit A,B cnh , cho nn AB c nh . V C chy trn (O;R) , D l nh ca C qua php tnh tintheo AB , cho nn D chy trn ng trn O l nh ca ng trn O- Cch xc nh (O) : T O k ng thng // vi AB , sau dng vc t OO' AB= .T O quay ng trn bn knh R , chnh l ng trn qu tch ca D.V d 3. Cho hai ng trn (O;R) v (O;R) cng vi hai im A,B . Tm im M
trn (O;R) v im M trn (OR) sao cho 'MM AB= .Giia. Gi s ta ly im M trn (O;R). Theo gi thit , th M l nh ca M qua php tnhtin theo vc tAB . Nhng do M chy trn (O;R) cho nn M chy trn ng trnnh ca (O;R) qua php tnh tin . Mt khc M chy trn (O;R) v th M l giaoca ng trn nh vi ng trn (O;R).
b/ Tng t : Nu ly M thuc ng trn (O;R) th ta tm c N trn (O;R) lgiao ca (O;R) vi ng trn nh ca (O;R) qua php tnh tin theo vc t ABc/ S nghim hnh bng s cc giao im ca hai ng trn nh vi hai ng trn
cho .V d 3. Cho ng trn (O) ng knh AB c nh . Mt ng knh MN thayi . Cc ng thng AM v AN ct cc tip tuyn ti B ln lt l P,Q . Tm qutch trc tm cc tam gic MPQ v NPQ ?
Gii- Tam gic MPQ c QA l mt ng cao , v vy nu ta k MM vung gc vi PQth MM ct QA ti trc tm H . OA l ng trung bnh ca tam gic MNH suy ra :
2MH OA BA= = . Vy php tnh tin theo BA bin im M thnh im H . Nhng Mchy trn (O;AB) cho nn H chy trn ng trn nh ca (O;AB) qua php tnh tin
BA .- Tng t i vi tam gic NPQ .- Gii hn qu tch . Do M khng trng vi A,B cho nn trn ng trn nh b i haiim nh ca A,B .
BI TON 2:TM IM M TRN NG THNG D SAO CHO KHONG CCH
MA+MB NGN NHT ( A,B- C NH CHO TRC )Cch gii
Bc 1: Tm im A i xng vi im A qua ng thng d . ( Khi ngthng d l ng trung trc ca AB , suy ra M thuc d th MA=MA ).

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d/ Vit phng trnh nh ca (H) :2 2
116 9
x y =
Giia/ Gi M(x;y) thuc cc ng cho v M(x;y) thuc cc ng nh ca chng.
Theo cng thc ta ca php tnh tin ta c :' 1 ' 1
' 2 ' 2
x x x x
y y y y
= + =
= + = + Thay x,y vo phng trnh cc ng ta c :- ng thng a : 3(x-1)-5(y+2)+1=0 3x-5y-12=0- ng thng b : 2(x-1)+(y+2)+100=0 hay : 2x+y+100=0
b/ ng trn (C) : ( ) ( ) ( )2 2
' 1 ' 2 4 ' 1 ' 2 1 0x y x y + + + + = hay : 2 2 6x 5 10 0x y y+ + + =
c/ ng (E) : ( ) ( ) ( ) ( )2 2 2 2
' 1 ' 2 1 21 1
9 4 9 4
x y x y + ++ = + =
d/ ng (H): ( ) ( ) ( ) ( )2 2 2 2
' 1 ' 2 1 21 1
16 9 16 9
x y x y + + = =
Bi tp v nh :Bi 1. Cho hai ng trn khng ng tm (O;R) v (O;R) v mt im A trn(O;R) . Xc nh im M trn (O;R) v dim N trn (O;R) sao cho MN OA= .Bi 2. ( Lm bi tp 4;5;6 HH11NC-trang 9)Bi 3. ( Lm bi tp : 2;3- HH11CB-trang 7 )
Gi Bi 1. V : :
OAMN OA T M N= uuur . Do N nm trn ng trn nh ca (O;R) . Mt
khc N li nm trn (O;R) do N l giao ca ng trn nh vi vi (O;R) . T
suy ra cch tm :- V ng trn tm A bn knh R , ng trn ny ct (O;R) ti N- K ng thng d qua N v song song vi OA , suy ra d ct (O;R) ti MBi 2.a/ Bi 4-trang 9-HH11NC.- V A,B c nh suy ra : AB U= .- T gi thit : ' 'MM MA MB MM MB MA AB+ = = = . Chng t : : 'ABT M Muuur .- Nhng M chy trn (O;R) cho nn M chy trn ng trn (O;R) l nh ca (O;R) .
b/ Bi 5.
- Ta ca M v N l :' '
1 1 1 2 2 2
' '
1 1 1 2 2 2
os sin os sin' ; 'sin os sin os
x x c y a x x c y aM Ny x y c b y x y c b
= + = +
= + + = + + - Khong cch d gia M,N v khong cch d gia MN .
Ta c : ( ) ( )2 2
2 1 2 1MN x x y y= +
( ) ( ) ( ) ( ) ( ) ( )2 2 2 22 2 2 2
2 1 2 1 2 1 2 1' ' os sin os sinM N x x c y y c x x y y = + + + = +
- Php F l php di hnh
- Khi :'
0 sin 0; os 1'
x x ac
y y b
= += = = = +
. y l cng thc ca php tnh tin .
c/ Bi 6.

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- Nu ( ) ( ) ( ) ( )1 : ; ' ; ; '; ' ' '; 'F M x y M y x N x y N y x th khong cch gia hai im MN
v MN l : ( ) ( ) ( ) ( )2 2 2 2
' ' ; ' ' ' 'MN x x y y M N y y x x= + = + + . Chng t
MN=MNcho nn chnh l php di hnh .- Nu : ( ) ( ) ( )2 : ( ; ) ' 2x; ; '; ' ' 2x '; 'F M x y M y N x y N y . Khi khong cch hai im l :
( ) ( ) ( ) ( )2 2 2 2
' ' ; ' ' 4 ' 'MN x x y y M N x x y y= + = + .- R rng : MN< MN : Do y khng phi l php di hnh v theo nh ngha :Php di hnh l php bin hnh bin hai im thnh hai im m khng lm thay ikhong cch gia chng .Bi 3.a/ Bi 2- trang 7.- T B v C k cc ng thng // vi AG . Sau t BB=CC=AG ( T gicBCCB l hnh bnh hnh )- A s trng vi G . Tam gic GBC l nh ca tam gic ABC qua php tnh tin theo
vc t AG .- Nu D l nh ca php tnh tin theo vc t AG th : DAG A D= phi trng vi G .b/ Bi 3-trang 7.
- Theo cng thc ta ca php tnh tin : ( )'
'
3 1 2' ' 2;7
5 2 7A
A
xA A
y
= == = + =
v ta ca
im ( )'
'
1 1 2' ' 2;3
1 2 3B
B
xB B
y
= = = = = + =
.
- Nu gi M(x;y) thuc ng thng d v M(x;y) thuc ng thng d : l nh cang thng d qua php tnh tin theo vc t v th theo cng thc ta c php tnh
tin ta c :' 1 ' 1
'' 2 ' 2
x x x xM
y y y y
= = + = + =
. Thay vo phng trnh ca d : (x+1)-2(y-
2)+3=0 . Hay d: x-2y+8=0 .
Bi 3. PHP I XNG TRC1. NH NGHA :* Cho ng thng d . Php bin mi im M thuc d thnh chnh n . Bin mi imM khng thuc d thnh im M sao cho d l ng trung trc ca MM , c gi l
php i xng qua ng thng d ( hay l php i xng trc ) . ng thng d gi l
trc i xng2. BIU THC TA CA PHP I XNG TRC
Ta chn ng thng d trng vi trc Ox . Vi mi im M(x;y) , gi M(x;y) l nh
ca M qua php i xng trc th :'
'
x x
y y
= =
( chnh l biu thc ta )
3. TNH CHTa/ Tnh cht 1: Php i xng trc bo ton khong cch gia hai im bt k .
b/ Tnh cht 2: Php i xng trc bin mt ng thng thnh mt ng thng ,bin mt on thng thnh mt on thng bng n , bin mt tam gic thnh mt tam
gic bng n , bin mt ng trn thnh mt ng trn c cng bn knh .4. TRC I XNG CA MT HNHnh ngha :

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNH* ng thng d gi l trc i xng ca hnh H nu php di xng qua d bin hnh Hthnh chnh n .5. NG DNG
BI TON 1. TM QU TCH CA MT IMBi ton :Cho hnh H v mt im A thuc hnh H thay i . Tm qu tch ca im M khi A
thay i .Cch gii .
Bc 1: Xt mt v tr bt k ca A v M . Sau d tm trn H c mt ngthng c nh l trung trc ca on thng AM ( Chnh l trc i xng ).
Nu A chy trn mt ng (C ) no , theo tnh cht ca php di xng trc ,th M chy trn ng (C) l nh ca (C ) qua php i xng trc .
V d 1. ( Bi 10-tr13-HH11NC ) .Cho hai im B,C c nh nm trn ng trn (O;R) v im A thay i trn ngtrn . Hy dng php i xng trc chng minh rng trc tm H nm trn mt
ng trn c nh .Gii
- V hnh . Gi H l giao ba ng cao ca tam gic ABC . Ko di AH ct (O;R) tiH . Ni CH- Chng minh IH=IH . Tht vy
Ta c : 'A BCH = ( Gc ni tip chn cung BH ).(1)
Mt khc : ( )2'
CH ABA BCH
CI AH
=
. T (1) v (2) suy ra : 'BCH BCH =
Chng t tam gic HCH l tam gic cn . Do BC vung gc vi HH , chng tBC l ng trung trc ca HH . Hay H v H i xng nhau qua BC . Cho nn khi Achy trn ng trn (O;R) th H cng chy trn (O;R) v H s chy trn ng trn(O;R) l nh ca ng trn (O;R) qua php i xng trc BC- Gii hn qu tch : Khi A trng vi B v C th tam gic ABC suy bin thnh ngthng . V th trn ng trn (O;R) b i 2 im l nh ca B,C .* Ch : Ta cn c cch khc chng minh H v H i xng nhau qua BC .- K AA ( l ng knh ca (O) ) suy ra BHCA l hnh bnh hnh , cho nn BC iqua trung im I ca AH .- AH song song vi BC ( v cng vung gc vi AH )
- T suy ra BC l ng trung bnh ca tam gic AHH C ngha l BC i quatrung im ca HH . Mt khc AH vung gc vi BC suy ra BC l trc i xng caHH , hay H v H i xng nhau qua BC.V d 2. Cho tam gic ABC c trc tm Ha/ Chng minh rng cc ng trn ngoi tip cc tam gic HAB,HBC,HCA c bnknh bng nhau
b/ Gi 1 2 3, ,O O O l tm cc ng trn ni trn . Chng minh rng ng trn i quaba im 1 2 3, ,O O O bng ng trn ngoi tip tam gic ABC .
Gii .a/ Gi s 1O l tm ca ng trn ngoi tip tam gic HBC , th theo bi taons ca vd 1 1O chnh l nh ca (O) qua php i xng trc BC . Cho nn bn knh ca chng

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHbng nhau . Tng t hai ng trn ngoi tip ca hai tam gic cn li c bn knhbng bn knh ca (O) .b/ Ta hon ton chng minh c 1 2 3, ,O O O l cc nh ca O qua php i xng trcBC,CA,AB . V vy bn knh cc ng trn ny bng nhau . Mt khc ta chng minhtam gic ABC bng tam gic 1 2 3O O O .
BI TON 2. TM IMCHO NG THNG d V HAI IM A,B . TM IM M THUC d SAOCHO MA+MB NH NHT. ( Khi A,B l hai im nm v mt pha ca d ),
MA MB T GI TR LNNHT( A,B nm v hai pha ca d )Cch gii :
Bc 1: Tm im A i xng vi A qua ng thng d Bc 2: Ni AB , ng thng ny ct d ti M . L im cn tm . Bc 3: Chng minh M l im duy nht .
V d 1. (Bi 9-tr13- HH11NC)
Cho gc nhn xOy v mt im A nm trong gc . Hy tm im B trn Ox ,im C trn Oy sao cho tam gic ABC c chu vi nh nht .
Gii .- Tm A i xng vi A qua Oy , B i xng vi A qua Ox- Ni AB ct Ox ti B , ct Oy ti C . chnh l hai im cn tm- Chng minh B,C l hai im duy nht cn tm .Tht vy : Do A i xng vi A qua Oy , cho nn CA=CA (1) . Mt khc : B ixng vi A qua Ox cho nn ta c BA=BB (2) . Gi P l chu vi tam gic ABC thP=CA+CB+BA =CA+CB+BB=AB ( do t (1) v (2) ).
V d 2: Cho ng thng d v hai im A,B nm cng pha vi d . Tm im M trnd sao cho MA+MB t gi tr nh nht ?Gii
- Tm im A i xng vi A qua d- Ni AB ct d ti M . M chnh l im cn tm .- Tht vy : V A i xng vi A qua d cho nn MA=MA (1). Do :MA+MB=MA+MB=AB .- Gi s tn ti M khc M thuc d th : MA+MB=MA+MB 'A B . Du bng chxy ra khi AMB thng hng . Ngha l M trng vi M .
V d 3. Cho ng thng d v hai im A,B ( nm v hai pha ca d ). Tm im Mtrn d sao cho MA MB t GTLN .
Gii .- Gi A l im i xng vi A qua d- Ni AB ct d ti M . M chnh l im cn tm .- Tht vy : ' 'MA MB MA MB A B = = . Gi s tn ti mt im M khc vi M trn
d , khi : ' ' ' ' ' 'M A M B M A M B A B = . Du bng ch xy ra khi MAB thnghng , ngha l M trng vi M.
V d 4 . Cho hai ng trn (O;R) v (O;R) v mt ng thng da/ Hy tm hai im M v M ln lt nm trn hai ng trn sao cho d l ngtrung trc ca on thng MM

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b/ Hy xc nh im I trn d sao cho tip tuyn IT vi (O;R) v tip tuyn IT vi(O;R) to thnh mt gc TIT nhn ng thng d l ng phn gic trong hocngoi .
GiiV hnh :
a/ Gi s M nm trn (O;R) v M nm trn (O;R) ta mn yu cu bi ton- V d l trung trc ca MM cho nn M nm trn ng trn (C) l nh ca ngtrn (O;R) qua php i xng trc d . Mt khc M li nm trn (O;R) do vy M lgiao ca (C) vi (O;R)- T suy ra cch tm :
Tm hai ng trn nh ca hai ng trn cho qua php i xngtrc d ( Ln lt l (C) v (C)
Hai ng trn ny ct hai ng trn cho ti 1 2,M M . Sau k haing thng d v d qua 1 2,M M ct (O;R) v (O;R) ti 1 2' ; 'M M
Cc im cn tm l ( )1 1'M M v ( )2 2'M Mb/ Nu MT v MT nhn d l phn gic trong hoc ngoi ca gc TIT th MT v MTi xng nhau qua d . T suy ra cch tm :- Gi d l nh ca MT qua php i xng d ngha l d l tip tuyn ca ng trn(C ) l nh ca (O;R) qua php i xng trc d. Mt khc d l tip tuyn ca (O;R) .Cho d l tip tuyn chung ca (C ) vi (O;R) . T ta suy ra cch tm M :
Tm (C ) l nh ca (O;R) qua php i xng trc d K d l tip tuyn chung ca (C ) v (O;R) . Khi d ct d ti M . Chnh l
im cn tm .
Tng t p dng cho (O;R)- S nghim hnh bng s giao im ca cc tip tuyn chung ct d .
BI TON :3TM IM I XNG VI IM QUA MT NG THNG
Bi ton : Cho im A(x;y) v mt ng thng d : ax+by+c=0 . Tm ta im Bi xng vi im A qua ng thng d ?Cch gii :
Bc 1: Gi B(x;y) l im i xng vi A qua d v H l trung im ca AB
th iu kin : ( )( )
. 0 12
ABUH d
=
Bc 2: Gii hai iu kin (1) v (2) suy ra ta ca BV d 1.Cho im M(2;3) tm ta im N i xng vi M qua ng thng d : y=x
Gii- Gi N(x;y) l im i xng vi M qua d v H l trung im ca MN th M,N i
xng nhau qua d th iu kin l :( )
( )
. 0 1
2
MN U
H d
=
- Ta c : ( ) ( )
2 32; 3 1;1 ;
2 2
x yMN x y U H
+ + = = =
uuuur ur
.

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- iu kin (*)( ) ( )
( )
2 .1 3 .1 05 2
3;22 3 1 32 2
x yx y y
Nx y x y x
+ = + = = = + + = + ==
V d 2.Cho im M(2;-3) . Tm nh ca im M qua php i xng trc d : y-2x=0
Gii- Gi N(x;y) l im i xng vi M qua d v H l trung im ca MN th M,N i
xng nhau qua d th iu kin l :( )
( )
. 0 1
2
MN U
H d
=
- Ta c : ( ) ( )2 3
2; 3 1;2 ;2 2
x yMN x y U H
+ = + = =
uuuur ur
.
- iu kin (*)
( ) ( )1
2 .1 3 .2 02 4 0 14 13
;2 3 5 14 3 32 2 3
x y yx y
Nx y y xx
+ + = =+ + = = + = + = =
BI TON :4CHO NG (C ) V NG THNG d HY VIT PHNG TRNH
NG (C) L NH CA (C ) QUA PHP I XNG TRC dCCH GII
Bc 1: Trn ng (C ) ly hai im A,B
Bc 2: Tm hai im A,B i xng vi A,B qua php i xng trc d
Bc 3: Vit phng trnh ng (C) i qua A,BV d 1: Cho ng thng d : x-2y-2=0 v ng thng d: y=x . Lp phng trnhng thng (m) i xng vi ng thng d qua ng thng d .
Gii
- Tm giao ca d v d bng A(x;y) l nghim ca h :2 2 0 2
0 2
x y x
x y y
= = = =
.A(-2;-2)
- Trn d ly im M (3;3) . Gi N(x;y ) l im i xng vi M qua d .Gi H l
trungim ca MN th iu kin M,N i xng nhau qua d l : ( )( )
. 0 12
MN UH d
=
(*)
- Ta c : ( ) ( )3 3
3; 3 2;1 ;2 2
x yMN x y U H
+ + = = =
uuuur ur
- iu kin (*)( ) ( )
( )
3 2 3 .1 02x 9 5
5; 13 32 7 12. 2 0
2 2
x yy x
Nx yx y y
+ =+ = = = + + = = =
.
- ng thng (m) l ng thng i qua AN c vc t ch phng l ( )7;1AN= , nn
(m) c phng trnh l : 2 2 7 12 07 1
x y x y+ += = .

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHV d 2. Cho hai ng thng d: 2x-y+2=0 ; d : x+3y-3=0 . Lp phng trnh ngthng (m) i xng vi ng thng d qua ng thng d .Gii- Tm ta im A l giao ca d vi d . Khi ta A l nghim ca h hai
phng trnh :
32x 2 0 3 87
;3 3 0 8 7 7
7
xy
Ax yy
= + = = + = =
- Trn ng thng d chn im M(0;2)- Tm ta im N i xng vi M qua ng thng d . Khi nu M,N i xng
nhau qua d th iu kin :( )
( )
. 0 1
2
MN U
H d
=
(*) Vi H l trung im ca MN , Ul vc
t ch phng ca d . Ta c : ( ) ( )2
; 2 3; 1 ;2 2
x yMN x y U H
+ = = =
uuuur ur
.
- iu kin (*)( ) 33x. 2 .1 0
3x- 2 3 15 ;23 0 1 5 53. 3 0
2 2 5
y xy
Nx yx y
y
= = = = + + =+ = =
- ng thng (m) =(AN) i qua3 1
;5 5
N =
v c vc t ch phng
( )6 33
; / / 2;1135 35
AN U = =
uuur ur
.
Do (m) :
3 1
5 5 0 11x 2 7 02 11
x yy
+ = = + = .
V d 3 . Cho ng trn (C ) : 2 2 4x 2 1 0x y y+ + + = v ng thng d : 2x-y+2=0.Hy vit phng trnh ca ng trn (C) l nh ca (C ) qua php i xng trc d .
GiiDo tnh cht ca php i xng trc bin (C ) thnh (C) c cng bn knh . Cho nn tach cn tm ta tm I ca (C) i xng vi tm I ca (C ) .Vy t gi thit ta c tm I ca (C ) c ta : I(2;-1) v R=2 .- Gi I(x;y ) l tm ca (C)H l trung im ca II , ( )1;2U= l vc t ch phng
ca ng thng d . I i xng vi I qua d th iu kin :( )
( )
'. 0 1
2
II U
H d
=
(*)
-Ta c : ( ) ( )2 1
' 2; 1 1;2 ;2 2
x yII x y U H
+ = + = =
uur ur
.
- iu kin (*)( ) ( )
( )
x-2 .1 1 .2 0x+ 0 1
' 3;32 12 9 0 12. 2 0
2 2
yy x
Ix yx y y
+ + == = = + + = = + =
- Vy (C): ( ) ( )2 2
3 3 4x y+ + = .

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V d 4. Cho (E) :2 2
19 4
x y+ = . V ng thng d : x+y-2=0 . Lp phng trnh (E) l
nh ca (E) qua php i xng trc d .Gii- V (E) ch ra ta cc nh ca trc ln : A(3;0) ,A(-3;0) v ta hai nh ca
trc nh : B(0;2) ;B(0;-2 )- Tm ta ca 4 nh ca hnh ch nht c s l nh ca 4 nh hnh ch nht c sca (E) cho . Bng cch gii cc bi ton nh nh trn , d dng tm c to ca O(2;2) l nh ca O(0;0) , M(4;5) l nh ca M(-3;-2 ). N(4;-1 ) l nh ca
N(3;-2) . P(0;-1) l nh ca P(3;2) v Q( 0;5) l nh ca Q(-3;2) .- p dng cch v (E) ta suy ra cch v ca (E) .* Ch : y l bi ton tng i kh , cha gp trong cc thi i hc , nhngly v d ny l m rng cho trng hp i xng trc . D ng (C ) cho lng g i chng na , ta ch cn s dng tt kin thc hc l c th gii c .
BI TP T LUYNBi 1. Gi m l ng phn gic ngoi ca gc A ca tam gic ABC . Chng minhrng vi mi im M trn m , chu vi tam gic MBC khng nh hn chu vi tam gicABCBi 2. Cho (E) vi hai tiu im 1 2,F F . Gi M l mt im nm trn (E) nhng khngnm trn ng thng 1 2F F v m l phn gic ngoi ti nh M ca tam gic M 1 2F F .Chng minh rng m ch ct (E) ti M duy nht ( ng thng m nh th gi l tiptuyn ca E ti M )Bi 3. Cho ng trn (C ) : 2 2 6x 2 1 0x y y+ + + = . Tm phng trnh ng trn (C)
qua php i xng trc d : x-y-0 .Bi 4 . Cho hai ng thng d : x-y+2=0 v d: 3x+4y-1=0 . Tm ng thng m lnh ca ng thng d qua php i xng trc l d .Bi 5. Cho ng thng d: x+y-2=0 v hai im A(-4;-3) ,B(2;-1) . Tm im M trnd sao cho MA+MB t gi tr nh nhtBi 6. Cho hai im A(4;3) v B(-2;0) . Tm trn ng thng d : x+y-2=0 im Msao cho MA MB t ga tr ln nht .Bi 7.( Bi 39-tr106-BTHH10NC)
Cho tam gic ABC c nh A4 7
;5 5
. Hai ng phn gic trong ca hai gc B v C
ln lt c phng trnh x-2y-1=0 v x+3y-1=0 . Vit phng trnh cnh BC ca tamgic .
GI CCH GIIBi 1. K ng phn gic ngoi ca gc A . Tm im C i xng vi C qua m . T ac : MB+MC=MB+MC 'BC . M BC=AB+AC . Suy ra MB+MC+BC
AB AC BC + + . chnh l iu phi chng minh .Bi 2. Gi s trc ln ca (E) l 2a , tc l M nm trn E khi : 1 2 2aMF MF+ = . Theocch chng minh bi 1 , nu M nm trn phn gic m th :
1 2 1 2
' ' 2aM F M F MF MF+ + =. Du bng ch xy ra khi M trng vi M . Vy nu Mkhc M th M khng nm trn E . Suy ra m ct E ti mt im duy nht ti M .

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHBi 3. ng trn (C ) c tm I(3;-1) v bn knh R=3 . Gi I l tm ca ng trn(C) . Nu I v I i xng nhau qua d th ta c h :
( )3 1 0
4 0' 0;43 1
4 402 2
x yx y x
Ix yx y y
+ = + = = = + = = = .
Vy ng trn (C): ( )22
4 9x y+ = i xng vi (C ) qua trc i xng d .Bi 4. Gi A l giao ca d v d th ta A l nghim ca h :
( )2 0 1
1;13x 4 1 0 1
x y xA
y y
+ = = = + = =
. Trn d ly im M(0;2) . Tm M(x;y) l nh ca
M qua php i xng trc d ( c ( )4; 3U= Khi ta M l nghim ca h :
( ) 334 3 2 04x 3 6 0 33 625 ' ;1 2 3x 4 3 0 6 25 253 4 1 0
2 2 25
x y xy
Mx y yy
= = + = = + + + =+ = =
.
Khi ng thng m i xng vi d qua d l ng thng AM i qua A(-1;1) c
vc t ch phng ( )8 19
' ; / / 8; 1925 25
AM U = =
uuuuur ur
suy ra (m) :1 1
8 19
x y+ = . Hay ng
thng (m) : 19x-8y+27=0.Bi 5. Tm ta A(x;y) i xng vi A(-4;-3) qua php i xng trc d: x+y-2=0
Suy ra h :( )
( )
4 3 01 0 5
' 5;64 3 11 0 62 02 2
x yx y x
Ax y x y y
+ + = + = = = + = =+ = .
Lp ng thng (AB) i qua A(5;6) c vc t ch phng ( ) ( )' 3; 7 / / 3;7A B U= = .Do (AB):
5 3
6 7
x tt R
y t
= + = +
. Vy M l giao ca (AB) vi d cho nn ta ca M l
nghim ca h :
9
105 323 23 3
6 7 ;10 10 10
2 03
10
tx t
y t x M
x yy
= = +
= + = = + = =
Bi 6. Tng t cch lm bi tp 5 , ta c to A(x;y) i xng vi A(4;3) qua d l
nghim ca h :( ) 33 3 0
0 3 32 ' ;4 33 0 3 2 22 0
2 2 2
x y xx yAx y
x yy
= = = = + + =+ = =
.

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ng thng (AB) i qua B(-2;0) c vc t ch phng : ( )7 3
' ; / / 7;32 2
A B U = =
uuuur ur
.
Do (AB):2 7
3
x tt R
y t
= + =
. im M cn tm l giao ca (AB) vi d , cho nn ta
M l nghim ca h :
2
52 7 4 4 63 ;
5 5 52 0
6
5
t
x ty t x M
x yy
== + = = =
+ = =
.
Bi 7. Tm ta hai im M,N ln lt l nh ca A qua php i xng trc l haing phn gic ca hai gc B v C , th M,N phi nm trn BC .T ng thng (BC) chnh l ng thng (MN) : y+1=0 .
Bi 4. PHP QUAY V PHP I XNG TM1. nh ngha php quay .* Trong mt phng cho im O c nh v gc lng gic khng i . Php binhnh bin im O thnh im O, bin im M khc O thnh im M sao choOM=OMv gc (OM;OM)= . c gi l php quay tm O gc quay l .2. nh l :Php quay l php di hnh .3. Php i xng tm .* nh ngha : Php i xng qua im O l mt php bin hnh , bin mi im Mthnh im M i xng vi M qua O , c ngha l : ' 0OM OM+ = .* K hiu v cc thut ng :Php i xng tm O k hiu : OD . Trong O l tm i xng*Biu thc ta :Trong mt phng ta cho im I(a;b) . Nu php i xng tm I bin im M(x;y)
thnh im M(x;y) th :' 2a
' 2
x x
y b y
= =
( chnh l biu thc ta ca php i
xng tm ) .* Tm i xng ca mt hnh : L im sao cho bin hnh H thnh chnh n
*Biu thc ta ca php quay c tm I(a;b) im M(x;y) , im M(x;y) v gcquay l :
Trong mt phng ta Oxy cho Q(I, ), vi I(a; b). Khi Q(I, ) bin im M (x; y)
thnh M(x; y) xc nh bi:
++=+=
cos)(sin)('
sin)(cos)('
byaxby
byaxax(IVb) ( Chng minh cho HS )
4. Cc ng dng ca php quay v i xng tm .
BI TON 1: BI TON QU TCH IM

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHBi ton : Cho hnh H v mt im M thay i trn ng (C ) ( thuc H ). Tm qutch ca im N khi M thay i .Cch gii :
Bc 1: Tm mt im I c nh sao cho I l trung im ca MN Bc 2: Da vo tnh cht ca php i xng tm I ta suy ra qu tch ca N
V d 1. ( bi ton 2-tr17-HH11NC).Cho ng trn (O;R) v hai im A,B c nh . Vi mi im M , ta xc nh imM sao cho 'MM MA MB= + . Tm qu tch im M khi im M chy trn (O;R) .
Gii- Gi I l trung im ca AB . Theo tnh cht ca vc t trung tuyn th :
2MA MB MI+ = , suy ra : ' 2MM MI= . C ngha l I l trung im ca MM- V A,B c nh , cho nn I c nh . Do : 'ID M M . Nhng M chy trn
(O;R) cho nn M l nh ca M qua php i xng tm I s chy trn ng
trn nh ca (O;R)- Cch xc nh (O;R) nh sau : Ni IO ko di , t IO=IO . Sau ly O lmtm , quay ng trn c bn knh R .
V d 2. ( Bi 17-tr19-HH11NC).Cho hai im B,C c nh trn ng trn (O;R)v mt im A thay i tren ngtrn . Hy dng php i xng tm chng minh rng trc tm H ca tam gicABC nm trn mt ng trn c nh . ( Hay : tm qu tch ca H khi A thay i ).
Gii- V hnh theo gi thit cho . Ni ng knh AM , tm v tr ca H . Ta thy CH
AB v MBAB suy ra CH//BM . Tng t BH//MC v t gic BHCM lhnh bnh hnh , do oa hai ng cho BC v MH ct nhau ti trung im I caBC .
- Do B,C c nh cho nn I c nh . Vy H l nh ca M qua php i xng tmI . Mt khc M chy trn (O;R) do H chy trn ng trn (O;R) l nh ca(O;R) qua php i xng tm I .
V d 3. ( Bi 34-tr10-BTHH11NC) .Cho ng thng a v mt im G khng nm trn a . Vi mi im A nm trn a tadng tam gic u ABC c tm l G. Tm qu tch hai im B v C khi A chy trna?
Gii- V hnh . T hnh v v tnh cht ca tam gic u ta thy gc 0120AGC AGB = = .
Nh vy php quay tm G vi gc quay 0120 = bin A thnh C v bin A thnh B .Nhng A chy trn d v th B v C chy trn ng thng d l nh ca d qua phpquay 0120 = .V d 4. ( Bi 35-tr10-BTHH11NC).Cho ng trn (O) v tam gic ABC . Mt im M thay i trn (O) . Gi 1M lim i xng vi M qua A, 2M l im i xng vi 1M qua B v 3M l im i
xng vi 2M qua C . Tm qu tch im 3M ?Gii .
- V hnh . T hnh v ta c : Do 1M , 2M i xng nhau qua B cho nn ( )1 2 1BM BM=

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- V 2M v 3M i xng nhau qua C cho nn : 2 3CM CM = (2) . T (1) v (2) chng tBC l ng trung bnh ca tam gic 1 2 3M M M , c ngha l BC// 1 3M M (3) .- Gi D l trung im ca M 3M th AD l ng trung bnh ca tam gic
1 3 1 3D / /MM M A M M (4) . T (3) v (4) suy ra AD//BC v t gic ABCD l hnh bnh
hnh . C ngha l D c nh. Nh vy : 3:DD M M . M M chy trn (O) cho nn 3M
Chy trn ng trn (O) l nh ca (O) qua php i xng tm D .
BI TON 2: DNG HNHHy tham kho mt vi v d sauV d 1. ( Bi ton 3-tr17-HH11NC)Cho hai ng trn (O;R) v (O;R) ct nhau ti hai im B,C . Hy dng mtng thng d i qua A v ct (O;R) v (O;R) ln lt ti M v N sao cho A ltrung im ca MN .
Gii
- Gi s ng thng d dng xong , do A l trung im ca MN cho nn N l nhca M qua php i xng tm A v vy N phi nm trn ng trn (O) l nh cang trn (O;R) ( v M chy trn (O) ). Mt khc N li thuc (O;R) v th cho nn
N l giao ca (O) vi (O;R) . T suy ra cch dng .+/ Dng ng trn (O) l nh ca ng trn (O) : Ni OA , t OA=OA .+/ ng trn (O) ct ng trn (O) ti N . Ni NA ct (O) ti M .
- Gii hn qu tch : S nghim hnh bng s giao im ca (O) ct (O) .V d 2. ( Bi 18-tr19-HH11NC)Cho ng trn (O;R) , ng thng d v im I . Tm im A trn (O;R) v im B
trn d sao cho I l trung im ca on thng AB .Gii- V hnh . Do I l trung im ca AB cho nn B l nh ca A qua php i xng tmI . Mt khc A chy trn (O;R) v th B chy trn ng trn (O) l nh ca (O) qua
php i xng tm I . Nhng B li nm trn d v vy B l giao ca d vi (O)-T suy ra cch tm . Ni IO t IO=IO , sau dng ng trn (O) bn knhR , ct d ti B . Ni BI ct (O;R) ti A .- Gii hn qu tch : S nghim hnh bng s giao im ca (O) vi d .
BI TON 3: BI TON CHNG MINH lm c dng bi ton chng minh ta cn phi lm chc kin thc v php ixng tm v php quay . ng thi phi nh li cc kin thc v tam gic , t gic :Hnh bnh hnh , hnh vung , hnh ch nht .V d 1. ( Bi ton 1-tr17-HH11NC)Cho hai tam gic u OAB v OAB . Gi C v D ln lt l trung im ca ccon thng AA v BB . Chng minh rng OCD l tam gic u ?
GiiXt php quay tm O vi gc quay bng gc lng gic ( OA,OB)= 060 . R rng A
bin thnh B v A bin thnh B , v th cho nn php quay bin on thng AAthnh on thng BB . T suy ra php quay bin C thnh D , do OC=OD .V gc quay bng 060 cho nn tam gic cn OCD l tam gic u .

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V d 2. ( Bi 43-tr11-BTHH11NC)V pha ngoi ca tam gic ABC v cc hnhvung BCMN v ACPQ c tm l O v O .a/ Chng minh rng khi c nh hai im A,B v cho C thay i th ng thng NQlun i qua mt im c nh .
b/ Gi I l trung im ca AB . Chng minh rng IOO l tam gic vung cn .
Gii .a/ V hnh theo gi thit cho . T hnh v , gii cho hc sinh bi ton ph : Cho haiim A,B c dnh , vi mi im M v vi hai php quay tm A , tm B c cng gcquay th php hp ca hai php quay l mt php i xng m tm i xng l nhgoc vung ca tam gic vung cn OAB ( O l tm i xng ).- Nh vy : : :A BQ C N Q C Q NQ i qua tm i xng H c xc nh bngcch dng tam gic vung cn HAB
b/ Tng t nh trn : ': ; :O OQ C B Q C A AB i qua tm i xng I c xcnh bng tam gic vung cn OOI ( vi I l nh ca gc vung ). Nh vy tam gic
OOI l tam gic vung cn . BI TON 4:TM NH CA MT HNH BNG PHP QUAY V PHP I XNG TM
CCH GII .S dng cc nh ngha , tnh cht ca php quay v php i xng tm cng vi biuthc ta ca chng .V d 1. ( Bi 1-tr15-HH11CB)Trong mt phng Oxy cho im A(-1;3) v ng thng d c phng trnh : x-2y+3=0 . Tm nh ca A v d qua php i xng tm O
Gii- Gi A(x;y) l nh ca A qua php i xng tm O(0;0) . Theo cng thc ta caphp i xng ta c :
( )' 0 ' ' 1
' 1; 3' 0 ' ' 3
x x x x xA
y y y y y
= = = = = = =
- Tng t Gi M(x;y) l mt im bt k thuc d v M(x;y) l mt im bt kthuc d l nh ca d qua php i xng tm O . Theo cng thc ta ca php i
xng ta c : ( ) ( )' 0 '
' 2 ' 3 0 ' 2 ' 3 0' 0 '
x x x xx y x y
y y y y
= = + = = = =
. Do d c
phng trnh l : x-2y-3=0 .V d 2. Trong mt phng Oxy cho ng trn (O;R) : 2 2 2x 6 6 0x y y+ + + = v (E) :2 2
19 4
x y+ = im I(1;2) . Tm nh ca (O;R) v (E) qua php i xng tm I
GiiGi M(x;y) l im bt k thuc (O;R) v (E) . T cng thc chuyn trc ta c :
( ) ( ) ( ) ( )
( ) ( )
2 2
2 2
2 ' 4 ' 2 2 ' 6 4 ' 6 0' 2.1 2 '
2 ' 4 '' 2.2 4 ' 1
9 4
x y x yx x x x
x yy y y y
+ + + == =
= = + =

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( ) ( )
2 2
2 2
6x 2 6 0
2 41
9 4
x y y
x y
+ + =
+ =
*Ch : (O;R) : ( ) ( )2 2
1 3 4 ( 1;6), 2x y J R+ + = = .Ta ch tm J(x;y) l nh ca J qua php i xng tm I(1;2) bng cng thc chuyn
trc ta : ( )' 2 ( 1) ' 3 ' 3;1' 4 (3) ' 1
x x Jy y
= = = = = .
Do (O) : ( ) ( )2 2
3 1 4x y + = l nh ca (O;R) qua php i xng tm I .
V d 3.( Bi 1.13-BTHH11CB)Trong mt phng Oxy cho ng thng d: x-2y+2=0 v d: x-2y-8=0 . Tm php ixng tm bin d thnh d v bin trc Ox thnh chnh n .
Gii tha mn yu cu bi ton th ta lm nh sau :
- Gi M(xy) thuc d , M(x;y) thuc d . Gi s tm i xng l I(a;b) , th theocng thc chuyn trc : ( ) ( )
' 2a2a 2 2 8 0 2 4 2a 8 0
' 2
x xx b y x y b
y b y
= = + + = =
.
- trc Ox thnh chnh n th tm i xng phi c dng : I(a;0) tc l b=0
- T hai kt qu trn ta c : ( )4 2a 8 2 3
3;00 0
b aI
b b
+ = = = = =
.
V d 4. ( Bi 1.14 tr-21-BTHH11CB)Cho ba im khng thng hng I,J,K . Hy dng tam gic ABC nhn I,J,K ln lt ltrung im ca cc cnh BC,CA,AB .
Gii- Phn tch : Gi s tam gic ABC dng xong tha mn iu kin u bi . V I,J,Kl trung dim cho nn l ng trung bnh suy ra =KB , tng t KJ=IC . T suyra cch dng :+/ Tm im P l nh ca J qua php i xng tm I+/ K Px //KJ v t PQ=KJ . T Q k Qy // v t QC=IP.+/ Tm B i xng vi C qua I v A i xng vi B qua K . Nh v tam gic ABC dng xong .* Ch : Ngoi cch trn ta cn c cch khc nh sau
+/ Ly mt im N bt k . Tm cc im M i xng vi N qua I , P i xng vi Nqua J v Q i xng vi P qua K . ( V hnh )+/ T suy ra : CM BN AP CQ= = = . Do C l trung im ca MQ . T suy racch dng .V d 5 ( Bi 1-tr19-HH11NC)Cho hnh vung ABCDa/ Tm nh ca im C qua php quay tm A gc quay 090
b/ Tm nh ca ng thng BC qua php quay tm O gc quay 090 .Gii
a/ T hnh v ta thy nh ca C qua php quay tm A gc 090 l C hoc C sao chocc tam gic ACC v ACC l cc tam gic vung cn

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHb/ Ta nhn thy nh ca C qua php quay tm O gc quay 090 l B hoc D . Cn nhca B qua php quay tm O gc quay 090 l A hoc C , do nh ca BC l AB hocDC .V d 6 .( Bi 2-tr19-HH11-NC) .Trong mt phng Oxy cho im A (2;0) v ng thng d : x+y-2=0 . Tm nh caim A v d qua php quay tm O gc quay 090 .
Gii- V hnh . T hnh v ta thy A thuc d . nh ca A qua php quay tm O gc quay
090 . L B(0;-2) hoc B(0;2) . im B c nh qua php quay l A(2;0) hoc A(-2;0)- V B v A nm trn d cho nn nh ca d qua php quay ny s l (AB) hoc (AB)
ln lt c phng trnh : 1 2 0; 1 2 02 2 2 2
x y x yx y x y
= = + = + = .
PHP V T
I. NH NGHA
Cho im O v mt s 0k . Php bin hnh bin mi im M thnh mt im M saocho 'OM kOM = c gi l php v t tm , t s v t l k .
K hiu : ( , ) : 'O kV M M , hay : M= ( ) ( ) ( ), 1,'
O kO
k
V M M V M
=
II. TNH CHT- Tnh cht 1. Nu php v t t s k bin hai im M,N thnh hai im M,N th :
' 'M N k MN= .- Tnh cht 2: Php v t t s k :a/ Bin ba im thng hng thnh 3 im thng hng v bo ton th t cc im y
b/ Bin mt ng thng thnh mt ng thng song song hoc trng vi ngthng , bin mt tia thnh mt tia , bin mt on thng thnh mt on thng .c/ Bin mt tam gic thnh mt tam gic ng dng vi n , bin mt gc thnh mtgc bng n .d/ Bin ng trn thnh ng trn c cng bn knh .III. TM V T CA HAI NG TRN1.nh l : Vi hai ng trn bt k lun c mt php v t bin ng trn nythnh ng trn kia v ngc li .Tm ca php v t gi l tm v t ca hai ngtrn2. Cch tm tm v t ca hai ng trn .
Trng hp : I trng vi I . Khi php v t tm I t s R/Rv php v t tmI t s -R/R bin ng trn (I;R) thnh ng trn (I;R) .
Trng hp I '; 'I R R . Trn (O;R) ly mt dim M bt k , trn (O;R) lyim M sao cho IM//IM v IM//IM . Hai ng thng MM v MM ctng thng ni hai tm II ti hai im O v O . Khi O nm ngoi II gi ltm v t ngoi , cn O nm trong on II gi l tm v t trong .
Trng hp I khc I v R=R . Khi MM//II nn ch c php v t tm Ovi k=-1 . chnh l php i xng .
IV. CC DNG TON THNG GP
BI TON 1 :TM NH CA MT HNH QUA MT PHP V T

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHS dng nh ngha v cc tnh cht ca php v t . T nh ngha nu tm v t lI(a;b) , im M(x;y) im M(x;y) th ta c :
( )
( )
( )
( )
' ''
' '
x a k x a x k x a aIM k IM
y b k y b y k y b b
= = + = = = +
uuuur uuur
(*) .
Chnh l biu thc ta ca php v t tm I t s v t l k .
.V d 1. Trong mt phng ta cho ng thng d: 3x+2y-6=0 . Hy vit phngtrnh ca ng thng d l nh ca ng thng d qua php v t tm I(1;2) t s vt k=-2 ?
GiiGi M(x;y) thuc d ,M(x;y) l mt im bt k thuc d th theo biu thc ta ca php v t ta c :
( )( )
' 1 ' 31
' 1 2 1 2 2' 2 ' 6' 2 2 2
22 2
x xx
x xy yy y
y
= + = = = = + =
.
Thay vo phng trnh ca ng thng d:' 3 ' 6
3 2 2 0 3x ' 2 ' 9 02 2
x yy
+ = + = Do vy d: 3x+2y-9=0 .V d 2 .( Bi 1.23-tr33-BTHH11CB)Trong mt phng Oxy cho ng thng d: 2x+y-4=0a/ Hy vit phng trnh ng thng d l nh ca ng thng d qua php v t
tm O t s v t k=3 .b/ Hy vit phng trnh ng thng d l nh ca d qua php v t tm I (-1;2) ts v t k=-2
Gii
a/ T cng thc ta :( )
( )
'' 0 3 0 ' '3 2 4 0 2 ' ' 12 0
' 3 3' 0 3 03
xxx x x y
x yyy y
y
= = + = + = = =
Do ng thng d: 2x+y-12=0 .
b/ Tng t :
( )
( )
' 1 ' 31' 1 2 1 ' 3 ' 62 2 2 4 0 2x ' ' 8 0
' 2 ' 6 2 2' 2 2 22
2 2
x xxx x x y
yy yy y
y
+ + = =+ = + + + = + + = = = + =
.
Do ng thng d: 2x+y+8=0 .V d 3. ( Bi 1.24-tr33-BTHH11-CB)Trong mt phng Oxy cho ng trn (C ): ( ) ( )
2 23 1 9x y + + = . Hy vit phng
trnh ng trn (C) l nh ca ng trn (C ) qua php v t tm I(1;2) t s k=-2 .Gii
ng trn (C ) c tm O(3;-1) bn knh R=3. Gi O (x;y) l tm ca (C) ,R lbn knh ca (C) . Ta c ta ca O tha mn biu thc ta ca php v t :Bin son : Nguyn nh S - T: 02403833608 Trang 19

20/28
( )
( )2 2
' 1 ' 31
2 2' 1 2 1' 2 ' 4 ' 3 ' 4
' 2 2 2 2 3 1 92 2 2 2
' ' 2.3 62
x xx
x xy y x y
y y y
R RR
= + = = = = + = + + = = = =
( ) ( )2 2
' 3 ' 6 36x y + + = . Vy (C) : ( ) ( )2 2
3 6 36x y + + =
BI TON 2:S DNG PHP V T GII CC BI TON HNH HC
xc nh mt im M ta xem n nh l nh ca mt im A no bit quaphp v t , hoc xem M nh l giao ca ca mt ng c nh vi nh ca mtng bit qua mt php v t .V d 1. Cho tam gic ABC c hai gc B,C u nhn . Dng hnh ch nht DEEG cEF=2DE vi hai nh D,E nm trn BC v hai nh F,G nm trn hai cnh AC v AB
. Gii- V hnh ( tha mn yu cu bi ton ).* Phn tch : + Gi s hnh ch nht dng xong , trn AB ly mt im G bt k ,dng hnh ch nht GFEF c EF=2DE v hai nh D,E thuc BC , ni BF ct
AC ti F , khi ta c :D 2
' D ' 2 ' ' ' '
BG G GF GF
BG G G F G F= = = . Chng t B,FF thng hng .Ta c
th xem hnh ch nht DEFG l nh ca hnh ch nht DEFG qua php v t tm B
t s v t :'
BGk
BG= . T suy ra cch dng .
* Cch dng :- Ly im G ty trn AB , sau dng hnh ch nht GFED c EF=2 DE,hai nh DE nm trn BC .- Ni BF ct AC ti F , ng thng qua F song song vi BC ct AB ti G . Gi D vE l hnh chiu ca G v F trn BC . Th hnh ch nht DEFG l hnh ch nht cndng .* Chng minh :
Tht vy : V GF //GF , GD//GD nn :D
.' ' ' ' '
GF BG G
G F BG G D= = T suy ra :
D ' ' 2' '
G G DGF G F
= = . Nh vy hnh ch nht dng tha mn yu cu bi ton .
V d 2. ( Bi 1.25-tr33-BTHH11CB).Cho na ng trn ng knh AB . Hy dng hnh vung c hai nh nm trnna ng trn , hai nh cn li nm trn ng knh AB ca na ng trn .
GiiV hnh , t hnh v ta c cc bc sau .* Phn tch .Gi s hnh vung MNPQ dng xong tha mn yu cu bi ton ( vi M,N nm
trn AB , cn P,Q nm trn na ng trn ).Gi O l trung im ca ABNi OQ v OP, dng hnh vung MNPQ sao cho M,N nm trn AB v O l trungim ca MN . Khi ta c :

21/28
' ' ' '
OQ OP PQk
OQ OP P Q = = = . Ta xem nh MNPQ l nh ca MNPQ qua php v t tm
O t s k=' '
PQ
P Q. T suy ra :
* Cch dng .- Dng hnh vung MNPQ ( c MN thuc AB v O l trung im ca MN )- Ni OP v OQ chng ct (O,AB) ti P v Q- Hnh chiu ca P v Q trn AB l N v M . Khi MNPQ chnh l hnh vung cndng .* Chng minh : Do MNPQ l hnh vung , cho nn MN//AB . Tam gic OMN
ng dng vi tam gic OPQ suy ra :' ' ' ' ' ' ' '
PQ OP OQ PN QMk
P Q OP OQ P N Q M= = = = = .
V d 3. ( Bi 1.26-tr33-BTHH11CB).Cho gc nhn Oxy v im C nm trong gc . Tm trn Oy mt im A sao chokhong cch t A n trc Ox = AC .
Gii- V hnh . Cn c vo hnh v ta c phn tch sau* Phn tch : Gi B l hnh chiu ca A trn Ox . theo u bi th tam gic ABC l tamgic cn nh A ( AB=AC ) . Gi s tam gic ABC l mt tam gic cn nh l A cAB vung gc vi Ox . D dng nhn thy hai tam gic ny ng dng v th ta c :
' ' ' ' '
AC OC ABk
A C OC A B= = = . Ta coi tam gic ABC l nh ca tam gic ABC qua php v
t tm OP t s v t l k . T suy ra cch dng :* Cch dng :
- Ni OC , sau trn Oy ly im A , tm B l hnh chiu ca A trn Ox ( k ABvung gc vi Ox) .- Dng com pa ly A lm tm , quay cung trn c bn knh bng AB ct OC ti C .- T C k hai ng thng song song vi hai cnh AC v CB chng ct hai cnhOy v Ox ti A v B . Tam gic ABC l tam gic cn tm* Chng minh : Ging cch phn tchV d 4. ( Bi tp O.11-tr76-BTHH10 T6-2000)Cho tam gic nhn ABC . Hy dng hnh vung MNPQ sao cho M,N nm trn cnhBC , P,Q nm trn hai cnh cn li ca tam gic .
Gii- V hnh . T hnh v ta c cch phn tch :Gi mt hnh vung MNPQ c cnh MN thuc BC v MN=NP=PQ=QMv bng a c nh Nu ta coi hnh vung MNPQ l nh ca mt php v t tm B vi
t s v t no th :' '
1' ' ' ' ' '
PQ PM PQ P QPQ PM
P Q P N PM P N= = = = . Suy ra cch dng .
- Trn AB ly mt im Q bt k , k ng thng qua Q vung gc vi BC ct BCti M . Sau t MN=AM , dng hnh vung MNPQ .- Ni BP ct AC ti P , k hai ng thng qua P // vi NP v MN chng ct BC
v AB ti N v Q . Cui cng k qua Q mt ng thng vung gc vi BC ct BC tiM ta c hnh vung MNPQ cn dng .* Ch : Ta cn c cch khc .

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNH- Dng hnh vung BCMN nm ngoi tam gic ABC . Gi BC ln lt l giao ca
AB v AC vi MN . Nh vy php v t tm A t s v t : k='
AB
ABbin tam gic
ABC thnh tam gic ABC , Cho nn bin hnh vung BCPQ thnh hnh vungMNPQ cn tm .V th ta ch cn k qua B v C hai ng thng vung gc vi BC chng ct cccnh Ac v AB ti cc im P v Q , ct BC ti N v M . Hnh vung MNPQ tm cV d 5. Gi A l giao hai ng ng trn ct nhau O v O Hy dng qua A mtng thng ct hai ng trn ti B v C sao cho AC=2AB
GiiV hnh minh ha . T hnh v ta c phn tch sau- T gi thit , ta c : 22 :AAC AB V B C
= . Nh vy php v t tm A bin Bthnh C . T ta c cch dng :- Dng ng trn nh ca ng trn (O) qua php v t tm A t s k=-2. Giao cang trn nh vi ng trn (O) l C . ng thng AC chnh l ng thng dcn dng .- Chng minh : Do C l nh ca B qua php v t tm A t s k=-2 cho nn AC=2AB .V d 6. Cho hai ng trn (O) v (O) tip xc ngoi vi nhau ti A( c bn knhkhc nhau ) .Mt im M nm trn ng trn (O) . Dng ng trn i qua M vtip xc vi O v O.
Gii- V hnh minh ha cho hc sinh . T c phn tch ..- Gi S l tm v t ngoi ca (O) v (O) ,N l nh ca M qua php v t tm S M lgiao im th hai ca AN vi (O) , Gi O l giao ca OM vi OM ( Ch :
OM//ON ) ta c : ( )'' '' '
' ' '' ' '
O M O M O N O M
O N M O= = nn OM=OM . Chng t (O) tip
xc vi (O) v (O) ti M v M .- Cch dng : Tm tm S ( k tip tuyn chung ca O v O ct OO ti S .
Ni SA ct (O) ti N v M. O chnh l giao ca OM vi OM .BI TON 3: QU TCH IM
gii mt bi ton qu tch im M khi im A thay i trn mt ng (C ) chosn . Trc ht ta cn phi lm mt s vic sau1. Trong hnh H cho , ta tm ra mt im A thay i trn mt ng (C ) cho sn
no ( c th l ng trn , c th l mt ng thng ) sao cho AM nm trn mtng thng i qua mt im c nh I no 2. Gn cho A v M cng vi I hai tam gic dng dng , t tm ra mt t s khngi k3. Vit ng thc vc t : IM k IA= kt lun M l nh ca A qua php v t tm Ivi t s v t l k .4. Nu A chy trn (C ) th M chy trn (C) l nh ca (C ) qua php v t tm I t sk . Nu cch dng (C) .V d 1. ( Bi 29-tr29-HH11NC) .
Cho ng trn (O;R) v mt im I c nh khc O . Mt im M thay i trnng trn . Tia phn gic gc MOI ct IM ti N . Tm qu tch im N .
Gii

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNH- V hnh . T hnh v v tnh cht ca ng phn gic trong chia cnh i din lmhai don thng t l vi hai cnh k ca hai cnh . Ta c kt qu sau :* Do O,I c nh cho nn OI=a khng i . Gi N l chn ng phn gic ca gc
MOI ( N thuc IM) , t ta c :NI OI a NI a a
IN IMNM OM R NM NI a R a R
= = = =+ + +
Hay :
a a
IN IM IN IMa R a R = =+ +
uur uuur
.V I c nh cho nn ( ), :I kV M N . Nhng M chy trn ng trn (O;R) cho nn Nchy trn ng trn (C) l nh ca (O;R) qua php v t tm I t s v t l k .* Cch xc nh (O;R) nh sau- Ni OI , tm O sao cho : O 'I kOI= , t suy ra O- Bn knh R c xc nh bng cng thc : k= R/R suy ra : R=kR .( Hoc : ly O lm tm quay mt ng trn c bn knh l ON )V d 2. ( Bi 8 N chng I-tr35-HH11-NC)Cho ng trn (O) c ng knh AB . Gi C l im i xng vi A qua B v PQl ng knh thay i ca (O)khc vi ng knh AB . ng thng CQ ct PA,PB ln lt ti M v N .a/ Chng minh Q l trung im ca CM , N l trung im ca CQ
b/ Tm qu tch ca cc im M,N khi ng knh PQ thay i .Gii
a. V hnh . T hnh x ta thy : Ni AQ, BQ , do C i xng vi A qua B cho nn tac B l trung im ca AC : BA=BC (1) . Mt khc BQ vung gc vi AQ ( gc nitip chn na ng trn ) PA vung gc vi AQ ( gc ni tip chn ng trn )suy ra PA // BQ do BQ l ng trung bnh ca tam gic ACM , ngha l Q l
trung im ca CM .- Tng t BN l ng trung bnh ca tam gic ACQ cho nn N l trung im caCQ : NC=NQ (2)
b/ T (1) v (2) ta c cc ng thc vc t :( ) ( );21 2 :CCM CQ V Q M = . Cho nn M chy trn ng trn (O) l nh ca (O)qua php v t tm C , t s v t bng 2 .
( ) 1.2
12 :
2 CCN CQ V Q N
= uuur uuur
. Vy N chy trn ng trn (O) l nh ca (O) qua
php v t tm C t s k=1/2 .- Hng dn hc sinh cch xc nh hai tm O v O.V d 3. ( Bi 9-tr35-HH11NC)Cho ng trn (O;R) v im A c nh . Mt dy cung thay i ca (O;R) c di bng m khng i . Tm qu tch cc im G sao cho 0GA GB GC+ + = .
Gi* V hnh cho hc sinh . T hnh v ly I l trung im ca BC , ni OI ( OI vunggc vi BC ) . A l mt im c nh ( c th nm trn (O) hay khng cn nm trn(O) . Do B,O c nh , gc OIB bng mt vung cho nn khi BC thay i I chy trn
ng trn tm O bn knh R=2
2
4mR . ( Xt tam gic vung BOI ).

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNH* T gi thit G l trng tm ca tam gic ABC . Theo tnh cht trng tm ca tam
gic ta c : 2;
3
2 2:
3 3 A
AGAG AI V I G
AI
= = uuur uur
. Do : G chy trn ng trn (O) l
nh ca ng trn (O;R) qua php v t tm A ,t s v t bng 2/3 .V d 4. ( Bi ton 6-tr39-HH11CB).
Cho tam gic ABC ni tip trong ng trn (O)bn knh R , cc dnh B,C c nhcn A thay i trn (O) .Chng minh rng trng tm G ca tam gic ABC chy trnmt ng trn .
Gii- V hnh , Gi I l trung im ca BC , th I c nh khi B,C c nh . Theo tnh cht
trng tm :1
31 1
:3 3 I
IG IA IG IA V A G= = uur uur
. Nhng A chy trn (O) do G chy
trn (O) l nh ca (O) qua php v t tm I t s 1/3.
- Xc nh (o;R) bng h :
1O ' O
13 ';1 3'
3
I I
O RR R
= =
uuur uur
V d 5. Cho hai ng trn (O;R) v (O;3R) tip xc trong vi nhau ti A. Nu Obin thnh O trong php v t tm A th t s v t bng bao nhiu ?
Gii- V hnh . T gi thit : AO=R, AO=R suy ra AO=3AO .Hay : 3' 3 : 'AAO OA V O O= . Do t s v t l k=3.
V d 6. Cho ng trn O v mt im P c nh ngoi (O) .T P k mt tip
tuyn thay i PBC . Tm qu tch trng tm G ca tam gic ABC ?Gii
V hnh . Gi I l trung im ca BC th theo tnh cht ca ng knh i qua imgia ca dy cung : OI vung gc vi BC . Nh vy I nm trn ng trn ng knhOP. Mt khc theo tnh cht trng tm , th G nm trn AI v cch A mt khong bng
2/3 AI , hay :2
32
:3 A
AG AI V I G= uuur uur
. Nhng I chy trn ng trn ng knh OP
cho nn G chy trn ng trn (O) l nh ca ng trn ng knh OP qua phpv t tm A t s 2/3.
- Cch xc nh O bng h :2'3
2'
3 2 3
AO AH
OP OP R
= = =
uuuur uuur
. ( Vi H l trung im ca OP )
V d 7. Cho ng trn (O;R) v mt im I c nh vi OI=2R . M l mt im ding trn O , phn gic gc IOM ct IM ti M . Tm qu tch im M khi M chytrn ng trn O.
Gii- V hnh . Theo tnh cht ca ng phn gic trong :
' 2 ' 2 2 ' 2 2 22 ' '' ' ' 2 1 3 3 3 3
M I OI R IM IM IM IM IM IMMM OM R IM M M IM
= = = = = = = =+ +
uuuur uuur

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHVy : Qua php v t tm I t s 2/3 bin im M thnh im M , nhng M chy trnng trn (O;R) cho nn M chy trn (O;R) l nh ca (O;R) qua php v t tm I .
- xc nh (O;R) :
2O ' O
32
'3
I I
R R
= =
uuur uur
.
V d 8. Cho hai ng trn (O) v (O) tip xc trong vi nhau ti A , ng knhk t A ct (O) ,(O) theo th t ti B,C . Qua A v ng thng d ct (O);(O) tiM,N . Tm qu tch giao im T ca BN v CM , khi d thay i ?
GiiV hnh minh ha .Cn c hnh v , ta c phn tch : BM v CN cng vung gc vi ng thng d , suyra BM//CN (1) . Hai tam gic OCN ng dng vi tam gic OBM cho nn :
2R ' ' ' '
2R '
TN CN CA R TN TB R R BN R R Rk BT BN
TB BM CB R BT R BT R R R
+ + += = = = = = = =
+
Hay : ' :'
R
R R
B
RBT BN V N T
R R+=
+
uuur uuur
. Nhng N chy trn (O;R) cho nn T chy trn
ng trn nh ca (O) qua php v t tm B t s k ='
R
R R+.
( HD hc sinh cch tm gii hn qu tch ) .
V d 9. ( Bi 73-tr17- n CI-BTHH11-NC).Cho im P nm trong ng trn (O). Mt ng thng thay i i qua P , ct (O)ti hai im A,B . Tm qu tch cc im M sao cho PM PA PB= + .
GiiV hnh minh ha choi hc sinh .Cn c hnh v ta c phn tch :- Gi I l trung im ca AB . Theo tnh cht trung im ca dy cung th OI vunggc vi AB , c ngha l I chy trn ng trn ng knh OP (1)- Theo quy tc vc t trung tuyn ta c : 2PM PA PB MI M= + = phi nm trn d doI,P nm trn d . V : PM=2MI=2(PM-PI) suy ra PM=2PI hay : 22 :PPM PI V I M= .Vy php v t tm P bin im I thnh thnh M . Nhng I li chy trn (O;OP) v thM phi chy trn ng trn nh ca (O) qua php v t tm P t s k=2.V d 10. Cho ng trn (O) v mt im P ngoi O . M l mt im thay i trnO . H l hnh chiu vung gc ca ca O trn PMa/ Tm qu tch trng tm G ca tam gic POM ?
b/ Tm qu tch cc im H v trung im I ca PH ?c/ Tm qu tch trng tm K ca tam gic OPH ?
Gii .V hnh minh ha cho hc sinh . T hnh v phn tch cho HS bit :-V H l hnh chiu ca O trn PM cho nn OH vung gc vi PM , cho nn H nmtrn ng trn O c ng knh OP .

26/28
CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNH- Gi J l trung im ca PO ( J l tm ng trn O) th G phi nm trn MJ v theo
tnh cht ca trng tm :1
31
:3
JJG JM V M G= uuur uuur
. Nhng M li chy trn ng trn O
cho nn G chy trn ng trn O l nh ca O qua php v t tm J t s k=1/3 .
- V I l trung im ca PH cho nn PI=1/2PH hay :1
21
:2 P
PI PH V H I= uur uuur
. Nhng H
li chy trn tm J bn knh2
OP, cho nn I chy trn ng trn nh ca ng trn
tm J qua php v t tm P t s k= .
- Trng tm K ca tam gic OPH phi nm trn JH v theo tnh cht trng tm , ta c :1
31
:3 J
JK JH V H K= uuur uuur
. Do vy K chy trn ng trn nh ca ng trn tm J bn
knh2
OPqua php v t tm J t s k=1/3 .
V d 11. Cho ng trn O v mt im A nm trong O , M l mt im di ngtrn ng trn O .a/ Tm qu tch trung im I ca AM ?
b/ ng trung trc AM ct ng trn O ti P v P . Tm qu tch chn ngvung gc H k t O n PP ?c/ Tm qu tch tm ng trn ngoi tip tam gic APP ?
Gii- V hnh minh ha cho hc snh . T hnh v hy ch cho hc sinh mt s kt qu :
* V I l trng im ca AM cho nn :
21 :2
AAI AM V M I= uur uuuur
. Nh vy qua php v ttm A t s bin M thnh I , nhng M chy trn ng trn O , cho nn I chytrn ng trn nh ca O qua php v t tm A t s k=1/2.* ng trung trc ca AM phi i qua I v vung gc vi AM .
BI TON 4. CHNG MINHTa hay gp bi ton chng minh mt ng thng i qua mt im c nh , hay mtim nm trn mt ng trn c nh , mt hnh vung tm li mt hnh H c nh
no . Khi ta ch cn chng minh ng thng i qua tm v t ca hai hnh Hv H hoc chng minh M nm trn mt ng trn nh ca mt hnh H qua mt phpv t tm I t s kV d 1. Cho hai ng trn (O 1 ) v ( 2O ) ngoi nhau , mt ng trn (O) thay itip xc ngoi vi ( )1 1;O R v tip xc ngoi vi ( )2 2;O R . Chng minh ng thngni hai tip im i qua mt im c nh .
GiiV hnh minh ha cho hc sinh . T hnh v , phn tch cho hc sinh thy :

27/28
* Gi M,N th t l hai tip im ca (O) vi hai ng trn ( )1 1;O R ; ( )2 2;O R . Th
1 2O O O O O = . K 2 1'/ /O M O M . Th ta c 1 2 'O M O M cho nn MM i qua tm v t
ngoi ca hai ng trn . Do : 2 21 1
' 'O M O N Rk
O M O M R= = = .
* Hai tam gic : ONM ng dng vi 2 'O NM suy ra :2
2 2 2
1' '
O NON OM ON ON OM O N O M OM O M
= = = = . Vy MN i qua tm v t ngoi c nh ca
hai ng trn : ( )1 1;O R ; ( )2 2;O R .
V d 2. Cho hai ng trn O v O tip xc ngoi vi nhau ti A .Mt gc vungxAy quay xung quanh A , tia Ax ct O ti M , tia Ay ct O ti M. Chng minhng thng MM lun i qua mt im c nh .
GiiNi MM ct O ti N ta thy : 'O N song song cng chiu vi AM . Tng t A lgiaoca OO vi vi O ta cng thy : ' ' / / / / ' 'A M AM OM O M
. Suy ra MM i qua
tm v t ca hai ng trn .V d 3. Cho tam gic nhn ABC vi trng tm G . Gi A,B,C ln lt l trungim ca cc cnh BC,CA,ABa/ Php v t no bin A thnh A,B thnh B v C thnh C ?
b/ Chng minh tm O ca ng trn ngoi tip tam gic ABC l trc tm tam gicABC
c/ Gi H l trc tm tam gic ABC , chng minh rng :1
2GO GH = uuur uuur
. Suy ra G,O,H
nm trn mt ng thng ( ng thng -le ) .Gii
a/ Theo tnh cht ca trng tm tam gic :1 1 1
2 2 21 1 1
' : '; ' : '; ' : '2 2 2G G G
GA GA V A A GA GA V A A GB GB V B B
= = = uuur uuur uuur uuur uuuur uuur
1
21
' : '2 G
GC GC V C C
= uuuur uuur
. Nh vy php v t tm G t s k=-1/2 bin ba im A,B,C
thnh ba im A,B,C .b/ V O l giao ba ng trung trc , cho nn OB AC , nhng AC//AC cho nnOBAC . Chng t OB l mt ng cao ca tam gic ABC .Tng t i vi OA v OC v vy O l trc tm ca tam gic ABC.c/ Do tam gic ABC l nh ca tam gic ABC qua php v t tm G t s k=-1/2 cho
nn H bin thnh O v :1
2GO GH = uuur uuur
.
BI TON 5.TM NH CA MT IM MT NG QUA PHP V T
* S dng ng thc vc t ca php v t v tnh cht bng nhau ca hai vc t , ta stm c kt qu .
V d 1. Trong mt phng ta Oxy , cho ng trn (O) : ( ) ( )2 21 1 4x y + = . Tmphng trnh ng trn (O) l nh ca (O) qua php v t tm O t s k=2 .

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CC DNG TON THNG GP TRONG PHP BIN HNH V PHP DI HNHGii
Tm I ca (O) c ta I(1;1) bn knh R=2 . Nu (O) c tm l J v bn knh R lnh ca (O) qua php v t tm O ta c ng thc vc t :
( )' 0 2.1 ' 2
OJ 2 2;2' 0 2.1 ' 2
x xOI J
y y
= = = = =
uur uur
. R=2R=2.2=4.
Vy (O) : ( ) ( )2 2
2 2 16x y + = V d 2. ( Bi 1.23-BTHH11-CB-tr33)Trong mt phng ta Oxy , cho ng thng d : 2x+y-4=0.a/ Vit phng trnh ca ng thng d l nh ca d qua php v t tm O t s k=3.
b/ Vit phng trnh ng thng d l nh ca d qua php v t tm I(-1;2) t s k=-2
Giia/Gi M(x;y) l mt im bt k thuc d v M(x;y) l nh ca M qua php v t tmO t s k=3 . Nu M chy trn d th M chy trn ng thng d .
Theo tnh cht ca php v t :'
' 3x 3' 3' 3 '
3
xxx
OM OM y y y
y
== = = =
uuuuur uuuur
.
Thay (x;y) vo d:' '
2 4 0 2x ' ' 12 03 3
x yy + = + =
. Vy d: 2x+y-12=0 .
b/ Tng t nh trn ta c :( )
( )
' 1 ' 31
' 1 2 1 2 2' 2
' 2 2 2 ' 2 ' 6
22 2
x xx
x xIM IM
y y y y
y
+ + = = + = + = =
= + =
uuuur uuur
.
Thay vo d :' 3 ' 6
2 4 0 2x ' ' 2 02 2
x yy
+ + = + + = . Do d: 2x+y+2=0 .
V d 3. ( Bi 1.24-tr33-BTHH11).Trong mt phng ta Oxy cho ng trn (C ): ( ) ( )
2 23 1 9x y + + = . Hy vit
phng trnh ng trn (C) l nh ca ng trn (C ) qua php v t tm I(1;2) ts k=-2.
Gii
Gi O(3;-1) l tm ca (C ) c bn knh R=3. ng trn (C) c tm J(x;y) bn knhR l nh ca (C ) qua php v t tm I t s k=-2 . Theo tnh cht ca php v t ta c :( )
( )( )
1 2 3 1 3IJ 2 O 3;8
82 2 1 2
x xI J
yy
= = = = = =
ur uur
. R=2R=2.3=6 .
Vy (C) : ( ) ( )2 2
3 8 36x y+ + = .

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