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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 .



12/28

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

= = = = + + =+ = =

.



13/28


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 .



16/28


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

+ + + == =

= = + =



17/28


( ) ( )

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



18/28

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



19/28

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



23/28

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 ).



24/28

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



25/28

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 .



28/28

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+ + = .

Cac Dang Toan Ve Phep Bien Hinh

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