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£^ TTpa^Eic;. H1 auro xpn 'MO"n"oil^E irpwrape9o5o KQI av elvai SuaKoAE^ 01 Trpd^Ei^ TOTE xPH^MOTroiqaE rn
(j£9o6o. Av eivai TrdiAi SuaKoAo ava^r]Tr|crE Tpfirj |JE0o5oEivai TTIO KOITOJ:
as
No pptGci n £$iauan TOU KuicAou TTOU E/EI icevrpo TO(3,4) KQI TTCpVti OTTO TO Cfr\\l£\O (0,5). Na PPE0EI
€ £cpaTTTOp£vr|^ TOU KUKAOU aio (0,5). Avaurrj TEJJVEI TOV d§ova TU>V x CTTO T vo Ppc6cidAAq^ ecpaTTTOfjevn^ TOU KUKAOU TTOU
OTTO TO T.
Auan:
Oi 5uo. I' auTti
|j£9o6oi oQrjyouv ae TroAu 5uaKoAE«; TTpdTTEpiTnwar) ava^r|TOU(j£ ciAAo TpoTro.
180
H QKTlva TOU KUKAOU R = V32+(4-5)2 = 32+(-l)2 = VlO
2 = 10 => x2-6x + 9 + y 2 -8y + 16= 10
x2 + y 2 - 6 x - 8y +15 = 0
H e^iowaq Tqq EcpaTrTO^Evq^ OTO (0,5) Eivai
x^ + yy! - 3(x+x^ - 4(y+y^ + 15 = 0
5 y - 3 x - 4 y - 2 0 + 15 = 0
y -3x = 5 (1)
H ecpaTTTOfJEvq TSJJVEI TOV d^ova TOOV x OTO T (-—,0)
£XW cp = co
M = AAT = — = 3. A2 = AKT = T => ^2 = — ~ —
3-^ ^_ A,, -X2 7 7 3
7 7
5 , 6
3 _ 6 - 7 A .5 ~ 7 + 61.
21 + 18A3 = 30-35A3
181
53A3 = 9
9y = —53
9_
53
53y = 9x+ 15
i i<± /V /V A 14
Na (3p£0£i n £$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 E^iawaswv (1), (2), (3) KCIIOTI: g = -3, f = -4, c = 23
rj g = -1, f = -2, c = 3
'Apa oi s^iawaeiq TWV KUKAWV srvai:
x2 + y2 - 6x - 8y + 23 = 0
x2 + y2 - 2x - 4y + 3 = 0.
TO
3. Na pp£iT£ ipv E^iCTWcrr) 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 KOIVOE^iowan 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 ^TOU^E, 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$) Eivai
KtK2 =
fJE OTI K^2 = RI + R2 apa oi KUKAOI ecpomrovTai
fia va ppou|j£ TO aquxfo ETracpr)^ AUOUJJE TO auarnM0 T(Ajv
TWV 5uo KU
18x-18y =
^n
185
(H y = x Eivai q Koivq ECpaTTTO(j£vn TCOV 6uo KUKAWV)
AVTIKaGlOTOUJJE Tq y = X QTOV KI KOI EXOUfJE
x2+ x2-16x +32 = 0 =^> x2-8x + 16 = 0 => x = 4
'Apa TO aqiJEio ETracprjg Eivai TO (4, 4).
5. Na PPEITE rig c iaibacig TCOV £q>cmro|j£vu)v TOU KUKAOUx2 + y2 - 10x + 2y = 0 TTOU civai TiapdAAnAcg irpog Tqv cuGeio5x+y+3 = 0.
Auaq:
Oi £(paTTTO^j£V£^ 6a EXOUV E iawaEi TH^ popcprjg y = Ax + p.TTapdAAnA£<; TTpog Tqv 5x + y + 3 = 0, TOTE TO A = -5.
6a E
TO aucrrriMci TCOV
y = -5x + p
x2 + y2-10x + 2y = 0
x2 + (-15x + P)2 - 10x + 2(-5x + P) = 0
26x2 -10(P + 2)x + p2 + 2p = 0 (1)
Auar|
Ha va EXEI n (1) MovaSiKQ Auan, TTPETTEI n SiaKpfvouaa va Eivaifan M£ MlSsv. (A = 0)
A = 100(P + 2)2 - 4(26)(P2 + 2p) = 0
A = p2 - 48p - 100 = 0 => p = 50 n p = -2
'ETai oi E^iawoEi^ TCXJV e<paTno|j£vcuv Eivai y = -5x + 50 KOI
y = -5x -2
186
AZKHZEIZ HA AYZH
1. Na pp£ii£ Tqv €£iou)aq TOU KUKAOU o oiroioq:I) EXEI KEVTPO TO Or\\*Z\O K(-2, 1) KOI TTEpvti OTTO TO A(2, 4)
11) EXEI 5i6|JETpo AB Lie A(8, 6) KOI B(4, -2)
(Air. x2+y2+4x-2y-20=0x2+y2-12x-4y+20=0)
2. Na pp£0ei n E^iawaq TOU KUKAOU o OTTOIO^ eivai-rrpos TOY KUKAO K: x2 + y2 + 4x - 6y - 3 = 0 KOI EXEI OKTIVO laq METO \i\ao Tq$ 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 OHM^IO 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 a^ovaTU>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 Q^OVUJV 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 £^a)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 Q^OVO TCJV X.
OTTO TO (0,3) KQI
10. Na pp£0£i q £^iacjar) 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 E^iacuaEiq TCUV KUKAuv TTOU Trcpvouv OTTO TOio (4,3) EcpdmovTai aiov d$ova Oy KOI EXOUV TO Kcvipairdvo) orr|v tuGeia 3x=2y.
(Arr. x2+y2-4x-6y+9=0,2 2 100 50 625 .
X +/—x y + = 0 )^
13. Na PPEITE THV e^iEu9£ia x-y+1=0,op6oya>via TOV KU
TOU KUKAOU TTOU EXEI TO KEvipo TOU air|VTTEpvd airo TO or\\it\o (3,3) KQI icpvei
x2+y2-8x+2y-2=0.
(ATT. x2+y2-4x-6y+12=0)
14. Na ppeQouv 01 E£iati>a£i£ TUJV KuicAcuv TTOU EcpdTTTOVTai TOUd£ova y'y QTO (0, 8) KOI OTTOKOTTTOUV OTTO TOV d$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 E^iatbaei^ 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. Na5EI^ETE OTI EipdTTTOVTai
18. a) Na pp£6£i n E^auar) 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£i^ETE 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) K^O.0), R1=2V5JK2(3P-1),R2= lO, p) (2,-4), (4,2), Y) 45°)
26. Na pp£iT£ Tqv E^iacjar) TOU KUKAOU TTOU mpvd OTTO TO KoivdariM^ia TU>V KUKAiov: x2+y2=8 KOI x2+y2-4x+6y+4=0 KOIE<pdTTTETai TOU d^ova 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 O^OVO 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)
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