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171

KYKAOZ 10Tl nPEHEl NA HMnPIZElZ

KYKAOZ

1. E£iau>or) KuicAou

(a) (x - a)2 + (y - (3)2 = R2 Kevipo (a, |3) aiaiva R.

(P) x2 + y2 + 2gx + 2fy + c = 0 KEvrpo (-g, -f), R= s + f 2 - c

(Y) Av o KU«AO$ £X£i KEvrpo Tfjv apxn TWV a^ovuv TOTE EXEI E^f2 2 2x + y = R KCM Trapa|j£TpiK£q x = R auv0 y = Rrip6

2.

z^ Ca.;&) ^, $ ~ $ +£

KUKAOU OTO ar\\\t'\o (Xi,yi)TOU.

XX! + yy, + g(x + ) + f(y + y^ + c = 0

3. 0£an TOU crnM£iou (x1f y1

(a) x?+y?+2gx1+2fy1 + c >0

(p) x? + yf + 2gx! + 2fy! + c =0

(Y) x? + y? + 2gxi + 2fyT + c < 0

TOV KUKAO x2+y2+2gx+2fy+c=0

TO (XT, yi) EKTO^ TOU KUKAOU

TO (XL yi) TTQVCO CTOV KUKAO

TO (Xi, y! ) TOU KUKAOU

4, AuvapnAeyETai

Z(x1 Trpog TOV KUKAO x2+y2+2gx+2fy+c=0AK(Z) = xf +yj +2gXj +2iy, +c.

172

5. 0£an KQI KUKAOU.

AvTiKa9ioroup£ Ti"|vKQI (3piaKou|j£

TOU KUKAOUp' (3a8|jou. Av A n SiaKpivouaa

(a) A > 0 <^> n EuGsia TE^VEI TOV KUKAO a£ 5uo

((3) A = 0 <^> H £U6£fa £CpO[TTT£Tai TOU KUKAOU.

(Y) A < 0 <=> q EuGEia QEV TEJJVEI TOV KUKAO.

6. EuvGiiKq Y'a va T£|JVOVTQI 5uo KUKAOI opOoyuwia (Kd0£Ta)

(a)2g1g2+2f1f2 = c1 + c2

8.

7. OEQEK; 5uo KUKAcuv

(a) R! - R2 j < 6 < RT + R2 TEjJVOVTCU

(P) Ri + R2 = 6 <=> ECpOTTTOVTOI

(Y) RI - R2 = 5 <=> ECpaTTTOVTQI

(5) RI + R2 < 5

R, - R 2 > 5

01 KUKAOI 5£V TEMVOVTQI

KUKAujv:

Kd0£ KUKAOC; TTOU TTEpvd airo Tqv TOjjrj TWV KUKAwvx2+y2-*-2g1x-«-2f1y+c1 = 0 KOI x2+y2+2g2x+2f2y+c2 = 0 £xei E iawan

^ x2+y2+2g1x+2f1y+c1+A(x2+y2+2g2x+2f2y+c2) = 0 OTTOU A

(zl

COADIi ID>J 0 = Z +

012903ox OJID

-I0DX11AD ID*

oiI C

:iDA3

IAI

M

010D]30n3 AUiO nOADIi ID13XDjd£J OdiA^X OJ_

:5oiiodi

'0 = Z + *£ - Anoi odiA3> 01 13X3 IDM ( [, 'g)g '(e (|,-)vnoil no^n)i 001 UornDi3 U

srinoxoPUD o '/ '6 ox

sa = s{£) - A) + S(D - x):*DAj3 UOOODJ S |jA3r1noiU5 H "B DAUND

'D) nodiA3>t noi 53A3rfADi3iAfiD 5u

30

L HEI3VOLJA

NO3ZHMZV HZAV HI VIJ HEI3VOUA

174

'ET0i r| E Icnoari IS KM efvai:

KM: y-2 = 2(x-1) = > y - 2 = 2 x - 2

Auw TO au

y=2x

y = 2x

=> 3x - 2 = 2xx = 2y = 3x -2

K(2, 4), R = KA=

'Eiai n E^iawan TOU KUKAOU Efvai:

(x-2)2 + (y-4)2 = 10

x2 -4x + 4 + y 2 - 8y+ 16= 10

x2 + y 2 - 4 x - 8 y + 10 = 0.

B' Tporro^:

H ^r)TOU|j£vn E^iacoarj EIVQI TH^

ETT£l6r| 5l£pXETCU OTTO TO (-1, 3)

'Exou|j£ 1 +9 -2g + 6f + c = 0

-2g + 6 f+c = -10 (1)

ETTEiSn 6i£pX£Tai OTTO TO (3, 1)

9 + 1 + 6 g + 2f + c = 0

6g + 2f + c = -10 (2)

ETT£i5rj TO KEVTpo (-g, -f) pplaKETai TTdvco aTqv y - 3x + 2 = 0

-f+3g + 2 = 0 (3)

x2+y2+2gx+2fx+c=0

175

Auoupe TO auorriM0 TWV rpiwv e iacuaewv (1), (2), (3).

(2)-(1) => 8g-4f = 0 => 2g=f

AvTiKa9iarouME QTH (3)

=> -2g + 3g + 2 = 0 => g = -2

c= 10

'Apa, x2 + y2 - 4x - 8y + 10 = 0

YTOAEEH 2

Av ae K0ieivai

Tf|veva dyvtaaro KaiTW KOKAou KC»va

euftefa^ 015 trpo$ TOVjje <rrnv

rj TrpOKOTTTOuaarj 5iaKpM>uaa A = 0.

?nv Cfrrdaraarj TOU I VT^U TOU KiTqv euScia KOI av efvai k&\c R TOTE £ ivar

Na &£ix6d OTI q £u8eia x - y - 3 = 0 eivai £cpamo|j£vn TOUKUKAOU x2 + y2 -2x -1 = 0 KQI va ppeQei TO

Auaq:

x - y - 3 = 0

x2 + y2 - 2x -1 = 0

=> = x-3

176

x2 + (x-3)2 - 2x - 1 = 0

x2 + x2 - 6x + 9 - 2x - 1 = 0 <=> 2x2 - 8x + 8 = 0

=> x2 - 4x + 4 = 0 ^> (x-2)2 = 0

H A = 0 apa n £u0£ia £(pdTTT£Tai TOU KUKAOU QTO OTIIJEIO

(x-2)2 = 0 => x = 2, y = -1

Na pp£0£i av q £u6eia 3x - 4y + 14 = 0 ccpdTTTETai TOU KUKAOUx2 + y2 + 4x + 6y - 3 = 0.

AUOT):

To KEVTpo TOU KUKAOU £ivai TO (-g, -f) = (-2, -3). H airoaTaan TOUKEVTPOU (-2, -3) OTTO Tqv £u6£ia 3x - 4y + 14 = 0 efvai:

3(-2) - 4(-3) + 14 ~6+12

/3 2+4 2

= = 45

H aKTiva TOU KUKAOU efvai

d = R ECpaTTTETQI

0 = gYOCH - 00k Y9£ + + 001,

•Q = V i3U3duL IDN o = 93 + *AGIO XY = A Alii 3r1nCHOi0D>IUAV "Q = £)d3ig UQI3U3 •£) + XY = A 5UAdorl 5Ui IDAIS

X(H- SA + ?x oY»n>i(O'O) OiOIiD ID13X

:5oJiodi ^v

:l_jonv

UXdu Alii 9110 3rinod3d> non o = 92 +x noi AmA3r1oiiiDd)3 10 Anoe3d£J

•g ID* Y Q^matf Uo?X£>DQ 1013 IDX iDi3d3<fc ojoao 01 gup oi tUo 01

00 '$ ^'X = X UAsrio (A

4-5lri Dj3Qn3 IDAIS UA3r1oiaiDd)3 U no 3rtn

one 3rfnod3d>Uomogs toimUJ ti0U)*o 30

£ HHI3VOLJA

178

-64A2 + 120A =

8A(15-8A) = Q = 0, A= —8

=> y = 0 KQI y = — x Eivcu oi8

£<pCITTTO|JEVCJV.

TCJV

B' T

TO ar]|j£io (xi , yi) OTO OTTOIO ECpdrrrrcTai n £u0£la. H^r)TOU|j£vr| E iacjan Eivai Tqq |jopcpr|c;

XXT + yy-, - 5(x+X!) - 3(y+y1)+25=0. EiTEiSr] 6i£px£Tai OTTO TO (0,0)

=> -5xi -3yi + 25 = 0 (1)

To (XT , y^) £TraAn9£U£i KOI Tqv E^iacoar] TOU KUKAOU SqAaSr)

x '+y ; -10x1-6y1+25-0 (2)

Auoup£ TO auornija TCJV E^iacjaECjv (1) KQI (2) KQI EXOUJJE

(25-3v,)2 , 25-3yi1 -- ±I^_ + _ 10 -

625-150v,

625 - 150V! + 9yJ + 25yf - 625 = 0

; - 150y1 =0

y1(34y1-150) = 0 =^> y1 = 0,

r,ayi = o, X1 = ^

|50 = 75

34 17

179

£cpaTTTO|j£vqg y = 0

25-V.a yi =

22517

5 5.17 17

-5(x+Xl)-3(y+yi) + 25 = 0

40x 75v

T7+17

40x 75v

40 25—)-3(y + —)_45 =

200 225

75y-85x-51y = 0

-45x = -24y ^> 8y = 15x

nopaTrjpnar|

I1 aurq ir|v TTEpiTTTOjan o SEUTEPOC; ipoirog5uaKoAoTEp£^ TTpaÊic;. H1 auro xpn 'MO"n"oilÊ irpwrape9o5o KQI av elvai SuaKoAE^ 01 TrpdÊi^ TOTE xPH^MOTroiqaE rn

(j£9o6o. Av eivai TrdiAi SuaKoAo ava^r]Tr|crE Tpfirj |JE0o5oEivai TTIO KOITOJ:

as

No pptGci n £$iauan TOU KuicAou TTOU E/EI icevrpo TO(3,4) KQI TTCpVti OTTO TO Cfr\\l£\O (0,5). Na PPE0EI

€ £cpaTTTOp£vr|^ TOU KUKAOU aio (0,5). Avaurrj TEJJVEI TOV d§ova TU>V x CTTO T vo Ppc6cidAAq^ ecpaTTTOfjevn^ TOU KUKAOU TTOU

OTTO TO T.

Auan:

Oi 5uo. I' auTti

|j£9o6oi oQrjyouv ae TroAu 5uaKoAE«; TTpdTTEpiTnwar) ava^r|TOU(j£ ciAAo TpoTro.

180

H QKTlva TOU KUKAOU R = V32+(4-5)2 = 32+(-l)2 = VlO

2 = 10 => x2-6x + 9 + y 2 -8y + 16= 10

x2 + y 2 - 6 x - 8y +15 = 0

H eîowaq Tqq EcpaTrTOÊvq^ OTO (0,5) Eivai

x^ + yy! - 3(x+x^ - 4(y+y^ + 15 = 0

5 y - 3 x - 4 y - 2 0 + 15 = 0

y -3x = 5 (1)

H ecpaTTTOfJEvq TSJJVEI TOV dôva TOOV x OTO T (-—,0)

£XW cp = co

M = AAT = — = 3. A2 = AKT = T => ^2 = — ~ —

3-^ ^_ A,, -X2 7 7 3

7 7

5 , 6

3 _ 6 - 7 A .5 ~ 7 + 61.

21 + 18A3 = 30-35A3

181

53A3 = 9

9y = —53

9_

53

53y = 9x+ 15

i i<± /V /V A 14

Na (3p£0£i n £$ioioar| TOU KUKAOU TTOU TT£pvd OTTO TO(1,1) KQI OTTO TO KOivd vr\[j£iQ Tiuv KUKAcuv x2 + y2 - 2x - 5 = 0KOI x2 + y2 - 3x + 4y - 4 = 0.

Alton:

H ^Tovvtvr} E^iaajarj efvai rrj^ (Jopcprjq x2+y2-2x-5+A(x2+y2-3x+4y-4)=0

i 6i£pX£Tdi OTTO TO (1,1) ^> 1+1-2-5+A(1+1-3+4-4) = 0

_5-A = 0 A = -5 =>

x2+y2-2x-5-5(x2+y2-3x+4y-4)=0

x2+y2-2x-5-5x2-5y2+15x-20y+20=0-4x2-4y2+13x-20y+15=0

4x2-4y2-13x+20y-15=0

AIKHZEIZ AYMENEZ

1. Na PPEGEI n E iacuaq TOU KUKAOU TTOU ECPOTTTETQI ir\q eu9£iac;3x-4y+17=0 KOI EXEI TO 1610 KEVTpo |J£ TOV KUKAO x2+y2-4x+6y-11=0.

Auar):

TO KEVTpO TOU KUKAOU EIVQI,

182

2g = -4

2f = 6K(2,-3)

H aTToaTaarj TOU KEVTPOU K OTTO TP|V EuGsia 3x-4y+17 = 0,dvai n aKTfva TOU KUKAOU

R =Ax, +BY l 3(2)-4(-3)

VA 2 +B 2

'ETOI n E^iacoan TOU KUKAOU dvai

2 2(x-2) + (y+3) = 7

35

5

y2-4x + 6y-36 =

2. Na pp£JT£ TI^ £^IOU)O£I^ TlOV KUICA(OV (K, R) TTOU TTEpVOUV OTTO TO

ariM^'o A(2,3), EXOUV QKTIVQ R = V2 Kai TOKA Eivai TrapaAAnAo |j£ Trjv £u0£i'a y - x = 0./

N >

* A Auan:

Oi TOJV KUKAoov 9a Elvai

x2+y2+2gx+2fy+c=0

flepvouv QTTO TO an(JEio A(2,3), ETCJI,

22+32+4g + 6f + c = 0 => 4g + 6f

'Exouv QKTiva R= V2 , ETOT

R2 = g2 + f2 - c ^> 2 = g2 + f2 - c,

(1)

(2)

To £u9uYpau.u.o TMHMQ( KA sivai TrapdAAnAo JJE Ti]v Eu9aay - x = 0, dpa n KAfarj AKA 9a EIVCU (an ME Tqv KAfan T%-

AKA —x 2 -x , 2 + g

= -1, (3)

183

AUOUIJE TO auaTriija TWV Tpiwv Eîawaswv (1), (2), (3) KCIIOTI: g = -3, f = -4, c = 23

rj g = -1, f = -2, c = 3

'Apa oi sîawaeiq TWV KUKAWV srvai:

x2 + y2 - 6x - 8y + 23 = 0

x2 + y2 - 2x - 4y + 3 = 0.

TO

3. Na pp£iT£ ipv EîCTWcrr) TOU KUKAOU TTOU £xei 5i6|j£Tpo Tqv KoivqXop5rj TWV KUKAWV x2+y2-2x+y-5 = 0 KQI x2+y2-2x+4y-8 = 0.

Auarj:

fia va ppoujjE THV KOivrj xopSrj AuoupE TO auoTrifJa TWVaswv TWV Quo KUKAwv.

AcpaipoupE TWV Quo KUKAWV

x2 + y2 - 2x + y - 5 = 0

x2 + y2 -2x + 4y - 8 = 0

H KOivq eivai:

fia va ppouue TO KOIVOEîowan TOU £voc; KUKAOU

3y = 3 => y = 1

A KOI B avTiKa0i0Toup£ rq y = 1

x2+12-2x+1-5=0 =^> x2-2x-3=0

=> Xi =-1, x2 = 3

'Apa A(-1, 1), B(3, 1)

PO K TOU KUKAOU TTOU ^TOUÊ, SIVQI TO fJ£00 TWV A KOI B.

184

H aiaiva R = KB = ^ (S- l )+(1- I ) = 2

'Apa n e^fawon TOU KUKAOU TTOU EXEI SidjjEipo rrjv AB Eivai

4. Na ppEire TO KEVipa KQI TI$ TWV KUKAIOV

K2: x2 + y2 - 18x + 2y + 32 = 0

auv£X£ia va SEI' ETE OTI oi 5uo KUKAOI EtpdTTTovTaiKOI PPEITE riq ouvT£Tay|J£V£g TOU aqpeiou

Auori:

O KU T EX£I KEVTPO (0, 8) KOI CtKTIVC( RI

O KUKAOC K2 £X£l KEVTpO (9,-1) KOI QKTIVa R2 =

H QTroaTaan TWV KEVTpcov (n 5ioiK£VTpo$) Eivai

KtK2 =

fJE OTI K^2 = RI + R2 apa oi KUKAOI ecpomrovTai

fia va ppou|j£ TO aquxfo ETracpr)^ AUOUJJE TO auarnM0 T(Ajv

TWV 5uo KU

18x-18y =

^n

185

(H y = x Eivai q Koivq ECpaTTTO(j£vn TCOV 6uo KUKAWV)

AVTIKaGlOTOUJJE Tq y = X QTOV KI KOI EXOUfJE

x2+ x2-16x +32 = 0 =^> x2-8x + 16 = 0 => x = 4

'Apa TO aqiJEio ETracprjg Eivai TO (4, 4).

5. Na PPEITE rig c iaibacig TCOV £q>cmro|j£vu)v TOU KUKAOUx2 + y2 - 10x + 2y = 0 TTOU civai TiapdAAnAcg irpog Tqv cuGeio5x+y+3 = 0.

Auaq:

Oi £(paTTTO^j£V£^ 6a EXOUV E iawaEi TH^ popcprjg y = Ax + p.TTapdAAnA£<; TTpog Tqv 5x + y + 3 = 0, TOTE TO A = -5.

6a E

TO aucrrriMci TCOV

y = -5x + p

x2 + y2-10x + 2y = 0

x2 + (-15x + P)2 - 10x + 2(-5x + P) = 0

26x2 -10(P + 2)x + p2 + 2p = 0 (1)

Auar|

Ha va EXEI n (1) MovaSiKQ Auan, TTPETTEI n SiaKpfvouaa va Eivaifan M£ MlSsv. (A = 0)

A = 100(P + 2)2 - 4(26)(P2 + 2p) = 0

A = p2 - 48p - 100 = 0 => p = 50 n p = -2

'ETai oi E^iawoEi^ TCXJV e<paTno|j£vcuv Eivai y = -5x + 50 KOI

y = -5x -2

186

AZKHZEIZ HA AYZH

1. Na pp£ii£ Tqv €£iou)aq TOU KUKAOU o oiroioq:I) EXEI KEVTPO TO Or\\*Z\O K(-2, 1) KOI TTEpvti OTTO TO A(2, 4)

11) EXEI 5i6|JETpo AB Lie A(8, 6) KOI B(4, -2)

(Air. x2+y2+4x-2y-20=0x2+y2-12x-4y+20=0)

2. Na pp£0ei n Eîawaq TOU KUKAOU o OTTOIO^ eivai-rrpos TOY KUKAO K: x2 + y2 + 4x - 6y - 3 = 0 KOI EXEI OKTIVO laq METO \i\ao Tq$ OKTlvag TOU K.

(Air. x2+y2+4x-6y+9=0)

3. Na pp£0Ei r| E iauian TOU KUKAOU o OTTOIO^ £X£i KEVTPO TO atiM£'°K(3, -2) KOI £(paTTT£Tai Ti]t EuBEia^ y = 3x -1.

(ATT. X2+y2-6x+4y+3=0)

4. AivETai o KUicAo^ (x + 2) + (y - 2) = 18 KQI TO aqpEio A(2,1). Na5EI^£T£ OTI TO A EIVGI EOOITEpIKO OHMÎO TOU KUKAOU KOI VQ PpEITE

Ti]v E iacaaq TH^ x°P6n? TOU KUKAOU TTOU EXEI |JEao TO A.

(ATT. 4x-y-7=0)

5. Na pp£9£i q £ iou>an TOU KUKAOU TOU OTTOIOU TO KEVTPOppiaKEiai <rrr|v EuGeia x - 2y + 2 = 0 KOI £<paiTT£Tai TOU aôvaTU>v y OTO anii£io A(0,3).

(An. x2+y2-8x-6y+9=0)

6. KUKAO£ ME KEVTPO K, TT£pvd OTTO TFJV apXH TUJV QÔVUJV O KOI

£q>dTTT£Tai rq^ cuGcia^ y = 2 QTO aqpEfo B(4, 2). Na ppeiTE TqvE iauiarj TOU. Etriaqg va SEI ETE OTI q £<paTrro|j£vq TOU OTOaqM£io A(8,-6) auTou Eivai KaGEiq arqv aKTiva KO.

(ATT. x2+y2-8x+6y=0)

187

7. AIVOVTQI 01 KUKAoi: (Ki): x2+y2-6x-4y+9=0(K2): x2+

a) Na ppfGouv Ta KEvrpa KOI 01 QKT?VE£ TOU$.P) Na Pp£9£i q E iacjaq TOU PI£IKOU TOU$ dgova.Y) Na pp£6£i q 9£ar| TCJV 5uo KUKACUV.5) Na ppcGei n £ iauan THS £<paTTTop£vn$ TOU KUKAOU (Ki) aro

aqpEio (3, 4).£) Na ppcBouv KQI 01 5uo € i(rd)a£i TCJV etpaTTTOpevwv TOU

KUKAOU (Kj) TTOU OYOVTOI TTpO^ OUT6v OTTO TQV OpXH TWV

a£6vu>v.

(ATT. a) Ki(3,2) Ri=2, K2(-1,2) R2=1,P) 8x = 5, y) Aev £x°uv Koivci ar||J£ia

12x5)y = = 0, y =

8. AIVETQI o KUKAo^ x2+y2-12x+6y+20=0. Na 5£i^£T£ OTI:a) To KEvrpo TTOU ppi<TK£Tai TTdvcj aTqv £u9£ia x-2y=12.P) To ar)M£io A(1, 2) £ivai £â)T£piKO or||j£io TOU KUicAou.Y) Eivai op9oYOJVioig JJE TOV KUKAo x2+y2-4x-6y-14=0.

9. Na pp£0£i n £?icru)(Tn TOU KUKAOU TTOU T£pv£i opGoyaJVia TOVKUKAO x+y-4x+6y-7=0 TT£pvd£(p6TTT£TQI TOU QÔVO TCJV X.

OTTO TO (0,3) KQI

10. Na pp£0£i q £îacjar) TOU KUKAOU TTOU TT£pvd onr6 TOA(4,1) T£(jv£i op9oyu>via TOV KUKAO x2+y2+4x+6y-11=0 KOI n£<paTTTop£vn GTTO ai\i*tio A Eivai TTapdAAqAn JJ£ Tqv £u6£ia2y+x=7.

(ATT. x2+y2-6x+2y+5=0)

11. Na 5£i4£T£ OTITOU 1'

£<panroij£vn TOU KUKAou x2+y2-6x+2yi-5=0£<pdnT£Tai KOI TOU KUKAou 5x +5y2-4=0.

188

12. Na ppeiiE Tig EîacuaEiq TCUV KUKAuv TTOU Trcpvouv OTTO TOio (4,3) EcpdmovTai aiov d$ova Oy KOI EXOUV TO Kcvipairdvo) orr|v tuGeia 3x=2y.

(Arr. x2+y2-4x-6y+9=0,2 2 100 50 625 .

X +/—x y + = 0 )^

13. Na PPEITE THV eîEu9£ia x-y+1=0,op6oya>via TOV KU

TOU KUKAOU TTOU EXEI TO KEvipo TOU air|VTTEpvd airo TO or\\it\o (3,3) KQI icpvei

x2+y2-8x+2y-2=0.

(ATT. x2+y2-4x-6y+12=0)

14. Na ppeQouv 01 E£iati>a£i£ TUJV KuicAcuv TTOU EcpdTTTOVTai TOUd£ova y'y QTO (0, 8) KOI OTTOKOTTTOUV OTTO TOV d$ova x'x xopSrj

12 |jovd5u)v.

(ATT. (x-10)2+(y-8)2=102, (x+10)2+(y-8)2=102)

15. Na PPEITE nq Eîatbaei^ Ttov EcpaTriojJEVwv TOU KUKAOUx2+y2-2x+4y=0 TTOU Eivai KOtGETE^ ainv tuGEia y = 2x-1.

(An. y = --x-4, y = --

16. Na ppEGei n OEQH Tu>v KUKAcuv: x2+y2-6x+8y+16=0 KOI x2+y2

(ATT. ECpaTTTOVTdl

17. AIVOVTOI 01 KUKAoi: x2+y2-2x-4y+1=0 KOI x2+y2+4x+4y-1=0. Na5EIÊTE OTI EipdTTTOVTai

18. a) Na pp£6£i n Eâuar) TOU KUKAOU TTOU TE^VEI opGoyiovia TOV

|

189

TO KUKAO x2+y2-2x-4y-2=0, TT£pvd OTTO Tnv apxn TWV a^oviovKQI ano TO crnM io (2,1).

P) Na ppeGei t] Tijjrj TOU K UKTTCK(x2+2y2)+(y-2x+1)(y+2x+3)=0 vairapicrrd KUKAO.

(Air.a =0, p) K=-5)

1 9. Aivrrai o KUKAo£ x2+y2-2x-8y-8=0. Na pp£6ouv 01TOU K£VTpOU, TO MHKO£ Tr| OKTIVa^ KOI TO

TTOU aTTOKOTTTEl OTTO TOV d$OVa TU)V X.

(Air. R=5, pov.)

20. l~ia TTora Ti|jr| TOU K n £u8eia y=2x+K £(pdnTETai TOU KUKAOUx2+y2-4x-1=0.

(Air. K= 1, K = -9)

21. 'Evas KUKAO£ (K) rrepvd crrro TO

£X£I TO KEVTpO TOU TTOVU)

O(0,0) KQI A(8,0) KQI

£U8£IQ y= — X.4

a) Na 5£i££TE OTI n e icruarj TOU pirop£i va rrdp£i TH pop<pnX2+y2-8x-6y=0.

P) Na PPEI'TE Tqv e iaaiar] TH^ £<paTTTO[j£vri$ TOU KUKAOU CTTOa|j£io B, av OB 5idM£Tpdg TOU.

(EfrTcroeigAE/1998)(An. 4x+3y-50=0)

22. Na pp£9ei q £ iau)ar| KUKAou, nou TTEpvd and rr\v apxn TCUVa£6vojv KOI OTTO iqv TOjjri TCOV KUKACOV:

x2+y2+8x-4y+6=0

(ATT. 5x2+5y2+10x-8y=0)

23. Na 5Eix8£i OTI TO Mn*oq KUKACUV:

190

x2+y2+ax+py+c=0

x2+y2+px+ay+c=0 EIVOI -(a-a + - 4c

24. AivETai o KuicAoq x2+y2=25 KCM TO ar|[J£io 1(7,1). Na ppeiiEHS Ka|JTruAi]£ ornv OTTOia ppioKETai o

TOU M, av £ [j£ao THS TM, KOI T or\\it\o TOU KUKAOU.

(An. (14-x)2+(2-y)2=25)

25. AivovTai 01 KUKAOI:

K2: x2+qi2-6x+2ip=0a) Na pptiT£ TO Kcvipa KOI nq QKTIVE^ TLJV 5uo KUKACUV.p) Na 5Ei£ET£ OTI 01 6uo KUKAOI TE^VOVTOI aE 5uo ar|M£ia A KQI B

KOI va PPEITE Tiq auvrETaYMCv^ Ttuv crr||j£iajv QUTOJV.Y) Na 5£iÊTE OTI TO EuGuYpaMMO Tpnjja AB Eivai 5iap£Tpo^ TOU

KUKAOU K2.

5) Na UTTOAOYJOTTE Tqv o^tia Y^via TTOU axHMaTi£ouv 01 £<pa-TCOV 50o KUKAwv aTO ar||J£io A.

(ATT.a) KÔ.0), R1=2V5JK2(3P-1),R2= lO, p) (2,-4), (4,2), Y) 45°)

26. Na pp£iT£ Tqv Eîacjar) TOU KUKAOU TTOU mpvd OTTO TO KoivdariMîa TU>V KUKAiov: x2+y2=8 KOI x2+y2-4x+6y+4=0 KOIE<pdTTTETai TOU dôva TUJV x.

(An.x2+y2-8x+12y+16=0)

27. AIVETOI o KUKAOC x2+y2=R2 KOI TO ariM^'o TOU T (Rauv6,Rr|M®)-H £<paTTTOM€Vr| TOU KUKAOU CTO T T£|JV£I TOV OÔVO Ox OTO A KOI

TOV Oy aTO B. Na Pp£ii£ THV E iauar) TH^ KO|JTTuAr|?ppicjK£Tai:

191

O) O Y€U>METplKO£ TOTTO£ TOU MECTOU M TOU OT.

P) o YEWjJETpiKOt TOTTOC TOU \iioov N TOU AB.Y) O Y£WJJ£TplKO£ TOTTOC Tq$ TtTOpTrj^ KOpU9lfc T TOU Op0OYU>-

viou OBfA.

(Art. 4x2+4y2=R2, JL. + J * ! , * + _ * ! )4x 4y x y

28. KuKAo^ M£ KEVTPO K(5, 4) TEpv£i TOV d^ova TIOV x ara A KOI BcbaiE AB = 6 KOI OA < OB (O q ap/q TU>V a^ovtov).

o) No 5£i^£T£ OTI q Egiaaiaq TOU KUKAOU Ei'vai x2+y2-10x-8y+16=0P) H y = x T£pv£i TOV KUKAO Ota T KOI A (OF < OA). Na 5ef£ET£OTI q AA eivai SiapEtpo^ TOU KUKAOU.

29. AIVETOI o KUKAo£ x2+y2 = 25 KOI TO

i) Na 5Eir££TE OTI TO M EI'VCN Eaumpiicon) Na pp£0£i n £{iau>0r] TH^ £u6£ia^ TTOU

£X£i M£ao TOU TO ar|ME*0 M.in) Na ppEGouv oi auvT£TOY|J£V£^ TU>V

KOI TOU KUKAOU.

M( — ,— ) .

TOU KuicAou.i q X°p5n n OTTOIO

(ATT. x-7y+25=0, A(-4,3), B(3,4))

30. ATTO TO aqpfi M(x1f y^ TOU KUKAOU K^ x2+y2=a2 <p£pvoup£ 5uo£cpaTTTO|j£V£^ MA KOI MB TrpO£ TOV KUicAo K2: x2+y2=p2. Av qXOp5q AB £q>OTTT£Tai aTov KUKAO Ka: x2+y2=Y2» va CEI' ETE OTI

31. O KUicAoq K! IIE KEVTPO (a1; pt) KOI OKTIVO R1

TOU KUKAOU K2 |J£ KEVTpO (O2, p2> KOI OKTIVO R2 . Av q

0TO KOIVO aqpEio TOU^ TTEpvd OTTO iqv opxq TU>Vva 5£i£T£ OTI

192

32. KUKAOC. [IE KEVTPO P KOI OKTl'VO R £<pdlTTETai E^WTEpIKO (J£

6uo KUKAoug x2+y2=4 KOI x2+y2-6x+8=0. Na SEI ETE OTI q

T£T|Jr|H£Vn X TOU KEVTPOU P I (TOU TO I |J£ — + 2 KOI OTI TO PJ

TTdva) OTqv KQMTTuAn y2=8(x-1)(x-2).

33. AIVOVTOI oi OMOKEVTPOI KUKAOI KI: x2+y2=R2 KOI K2: x2+y2=2R2.ATTO £va CTHPEIO A(a, p) TOU K2 (pcpoupE TIC. EcpaTrropEVEc. TOUKUKAOU KV Na SEI ETE OTI 01 E<panTop€V£q OUTEC, EIVOI Kd0ET£q.

34. Na pp£ii£ Tqv E iawan TOU KUKAOU TTOU TTEpva OTTO TO ar]\itioA(-1, 2), £X£I TO K£VTpO TOU 0Tr]V EU0EIQ X = -3y KOI T£|JV£I Op9o-

TOV KUKAO x2 + y2 + 8x + 10y + 9 = 0.

(Air. 6x-2y+5 =

35. Na PPEITE THVA(4,2) KOI £cpdiTT£Ta

TOU KUKAOU TTOU TTEpva airo TO ar\\itioy = x aro ar\\izio B(1,1).

(An.

36. Na Pp£IT£ TIC, ££iaUKJ£l$ Tb>V KUKAOJV TTOU £XOUV TO KEVTpO TOUq

TTdvu) crrnv Eu0Eia y = x, TTEpvouv OTTO TO atiH^io (1,8) icaiairoKOTTTouv OTTO TOV d^ova Ttov x Tprjpa irjKou^ 6 |jovd6wv.

(ATT. x2 + y2 -28x -28y + 187 = 0x2 + y2-8x-8y + 7 = 0)

T

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