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"dinlsa`udirpayslks.ln" 9

I klrk bj ls i bhkus hf i phlnt ln i pbinj wahsj olstinkj frhd i flxjo phlnt (kibbjo kjntrj) ls ibwiys

khnstint (kibbjo riolus).

Jquitlhn hf i klrkbj ln virlhus fhrds(i) _aj klrkbj wlta kjntrj is hrlcln & riolus r ais taj jquitlhn0 x7 + y7 4 r7.

(g) _aj klrkbj wlta kjntrj (a, `) & riolus r ais taj jquitlhn0 (x a)7 + (y` )7 4 r7.

(k) _aj cjnjrib jquitlhn hf i klrkbj ls

x7 + y7 + 7cx + 7fy + k 4 =

wlta kjntrj is (c,f) & riolus 4 kfc 77 .

_als kin gj hgtilnjo frhd taj jquitlhn

(x a)7 + (y `)7 4 r7

x7 + y7 7ax 7`y + a7 + `7 r7 4 =_i`j a 4 c, ` 4 f ino a7 + `7 r7 4 k

Khnoltlhn th ojflnj klrkbj 5-

c + f k > = rjib klrkbj.

c + f k 4 = phlnt klrkbj.c + f k 2 = ldiclniry klrkbj, wlta rjib kjntrj, tait ls ( c, f)

Nhtj 5 _ait jvjry sjkhno ojcrjj jquitlhn ln x & y, ln walka khjfflkljnt hf x7 ls jquib th khjfflkljnt hf y7 & taj

khjfflkljnt hf xy ls zjrh, ibwiys rjprjsjnts i klrkbj.

(o) _aj jquitlhn hf klrkbj wlta (x9, y

9) & (x

7, y

7) is jxtrjdjtljs hf lts olidjtjr ls5

(x x9) (x x

7) + (y y

9) (y y

7) 4 =.

Klrkbj

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"dinlsa`udirpayslks.ln" 7

_als ls hgtilnjo gy taj fikt tait incbj ln i sjdlklrkbj ls i rlcat incbj.

([bhpj hf ]I) ([bhpj hf ]G) 4 9

9

9

xx

yy

.7

7

xx

yy

4 9

(x x9) (x x

7) + (y y

9) (y y

7) 4 =

Nhtj tait tals wlbb gj taj klrkbj hf bjist riolus pisslnc tarhuca (x9, y

9) & (x

7, y

7).

Jxidpbj # 9 5 Flno taj jquitlhn hf taj klrkbj wahsj kjntrj ls (9, 7) ino riolus ls ?.

[hbutlhn 5 _aj jquitlhn hf taj klrkbj ls (x 9)7 + (y (7))7 4 ?7

(x 9)7 + (y + 7)7 4 96 x7 + y7 7x + ?y 99 4 = Ins.

Jxidpbj # 7 5 Flno taj jquitlhn hf taj klrkbj walka pissjs tarhuca taj phlnt hf lntjrsjktlhn hf taj blnjs ;x

7y 9 4 = ino ?x + y 73 4 = ino wahsj kjntrj ls (7, ;).

[hbutlhn 5 Bjt ] gj taj phlnt hf lntjrsjktlhn hf taj blnjs IG ino BD wahsj jquitlhns irj rjspjktlvjby

;x 7y 9 4 = ..........(l)

ino ?x + y 73 4 = ..........(ll)[hbvlnc (l) ino (ll), wj cjt x 4 :, y 4 3. [h, khhrolnitjs hf ] irj (:, 3). Bjt K(7, ;) gj taj

kjntrj hf taj klrkbj. [lnkj taj klrkbj pissjs tarhuca ], tajrjfhrj

K] 4 riolus 77 );3()7:(

riolus 4 9=< .Ajnkj taj jquitlhn hf taj rjqulrjo klrkbj ls

(x 7)7 + (y + ;)7 4 79= k klrkbj kuts taj x ixls it twh olstlnkt phlnts.

c7 4 k klrkbj thukajs taj xixls.

c7 2 k klrkbj bljs khdpbjtjby ighvj hr gjbhw taj xixls.

IG 4 7IO 4 7 77 KOr 4 7 77 fr 4 7 7777 fkfc 4 7 kc7

Jxidpbj # : Flno taj jquitlhn th taj klrkbj thukalnc taj njcitlvj y-ixls it i olstinkj ; frhd taj hrlcln ino

lntjrkjptlnc i bjncta 1 hn taj x-ixls.

[hbutlhn 5 Bjt taj jquitlhn hf taj klrkbj gj x7 + y7 + 7cx + 7fy + k 4 =. [lnkj lt thukajs y-ixls it (=, ;)

ino (=, ;) bljs hn taj klrkbj.

k 4 f7 ...(l) < 6f + k 4 = .......(ll)Frhd (l) ino (ll), wj cjt < 6f + f7 4 = (f ;)7 4 = f 4 ;.]uttlnc f 4 ; ln (l) wj hgtiln k 4

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Jxidpbj # 6 5 Flno taj piridjtrlk jquitlhns hf taj klrkbj x7 + y7 ?x 7y + 9 4 =

[hbutlhn 5 Sj aivj 5 x7 + y7 ?x 7y + 9 4 = (x7 ?x ) + (y7 7y) 4 9 (x 7)7 + (y 9)7 4 77

[h, taj piridjtrlk jquitlhns hf tals klrkbj irj

x 4 7 + 7 khs , y 4 9 + 7 sln.

Jxidpbj # 3 5 Flno taj jquitlhns hf taj fhbbhwlnc kurvjs ln kirtjslin fhrd. Ibsh, flno taj kjntrj ino riolus hf

taj klrkbj x 4 i + k khs, y 4 g + k sln

[hbutlhn 5 Sj aivj 5 x 4 i + k khs, y 4 g + k sln khs 4k

ix, sln 4

k

gy

7

k

ix

+

7

k

gy

4 khs7 + sln7 (x i)7 + (y g)7 4 k7

Kbjirby, lt ls i klrkbj wlta kjntrj it (i, g) ino riolus k.

[jbf priktlkj prhgbjds 5

(6) Flno taj piridjtrlk jquitlhns hf klrkbj x7 + y7 6x + ?y 97 4 =

(3) Flno taj kirtjslin jquitlhns hf taj kurvj x 4 7 + ; khs, y 4 ; + ; sln

Inswjrs 5 (6) x 4 ; + : khs, y 4 7 + : sln (3) (x + 7)7 + (y ;)7 4 =.

Nhtj 5 _aj crjitjst & taj bjist olstinkj hf i phlnt I (bljs hutsloj taj klrkbj) frhd i klrkbj wlta kjntrj K &

riolus r ls IK + r & IK r rjspjktlvjby.

Jxidpbj # 1 5 Olskuss taj phsltlhn hf taj phlnts (9, 7) ino (6, =) wlta rjspjkt th taj klrkbj

x7 + y7 ?x + 7y 99 4 =

[hbutlhn 5 Sj aivj x7 + y7 ?x + 7y 99 4 = hr [ 4 =, wajrj [ 4 x7 + y7 ?x + 7y 99.

Fhr taj phlnt (9, 7), wj aivj [94 97 + 77 ? 9 +7 7 99 2 =

Fhr taj phlnt (6, =), wj aivj [74 67 + =7 ? 6 +7 = 99 > =

Ajnkj, taj phlnt (9, 7) bljs lnsloj taj klrkbj ino taj phlnt (6, =) bljs hutsloj taj klrkbj.

[jbf priktlkj prhgbjd 5

(1) A hw i rj t a j ph ln ts ( = , 9 ) ( ;, 9 ) i n o (9 , ; ) s lt ui tj o w lt a r js pj k t th t aj k l rk b j

x7 + y7 7x ?y + ; 4 =8

Inswjr 5 (1) (=, 9) bljs hn taj klrkbj 0 (;, 9) bljs hutsloj taj klrkbj 0 (9, ;) bljs lnsloj taj klrkbj.

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"dinlsa`udirpayslks.ln" :

Blnj ino i klrkbjBjt B 4 = gj i blnj & [ 4 = gj i klrkbj. Lf r ls taj riolus hf taj klrkbj & p ls taj bjncta hf taj pjrpjnolkubir

frhd taj kjntrj hn taj blnj, tajn5

(l) p > r taj blnj ohjs nht djjt taj klrkbj l. j. pissjs hut sloj taj klrkbj.(ll) p 4 r taj blnj thukajs taj klrkbj. (Lt ls tincjnt th taj klrkbj)(lll) p 2 r taj blnj ls i sjkint hf taj klrkbj.(lv) p 4 = taj blnj ls i olidjtjr hf taj klrkbj.Ibsh, lf y 4 dx + k ls blnj ino x7 + y7 4 i7 ls klrkbj tajn

(l) k7 2 i7 (9 + d7) taj blnj ls i sjkint hf taj klrkbj.(ll) k7 4 i7 (9 + d7) taj blnj thukajs taj klrkbj. (Lt ls tincjnt th taj klrkbj)(lll) k7 > i7 (9 + d7) taj blnj ohjs nht djjt taj klrkbj l. j. pissjs hut sloj taj klrkbj.

_ajsj khnoltlhns kin ibsh gj hgtilnjo gy shbvlnc y 4 dx + k wlta x7 + y7 4 i7 ino di`lnc taj olskrldlnint

hf taj quioritlk crjitjr tain zjrh fhr sjkint, jquib th zjrh fhr tincjnt ino bjss taj zjrh fhr taj bist kisj.

Jxidpbj # < 5 Fhr wait vibuj hf k wlbb taj blnj y 4 7x + k gj i tincjnt th taj klrkbj x 7 + y7 4 : 8

[hbutlhn 5 Sj aivj 5 y 4 7x + k hr 7x y + k 4 = ......(l) ino x7 + y7 4 : ........(ll)

Lf taj blnj (l) thukajs taj klrkbj (ll), tajn

bjncta hf taj frhd taj kjntrj (=, =) 4 riolus hf klrkbj (ll)

77 )9(7

k==7

4 :

:

k4 :

:

k4 : k 4 :

Ajnkj, taj blnj (l) thukajs taj klrkbj (ll) fhr k 4 :

[jbf priktlkj prhgbjd 5

(

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"dinlsa`udirpayslks.ln" 6

Nhtj 5 Ln cjnjrib taj jquitlhn hf tincjnt th iny sjkhno ojcrjj kurvj it phlnt (x9,

y9) hn lt kin gj hgtilnjo gy

rjpbiklnc x7 gy x x9,

y7 gy yy9,

x gy7

xx 9, y gy

7

yy 9,

xy gy7

yxyx 99 ino k rjdilns is k.

]iridjtrlk fhrd hf tincjnt_aj jquitlhn hf i tincjnt th klrkbj x 7 + y7 4 i7 it (ikhs , i sln ) lsx khs + y sln 4 i.

NH_J 5 _aj phlnt hf lntjrsjktlhn hf taj tincjnts it taj phlnts ]() & Y() ls

7

7

7

7

khs

slni,

khs

khsi

Jxidpbj#9=5 Flno taj jquitlhn hf taj tincjnt th taj klrkbj x7 + y7 ;=x + 6y + 9=< 4 = it (?, 9).

[hbutlhn 5 Jquitlhn hf tincjnt ls

?x + (y) ;=

7

?x+ 6

7

)9(y+ 9=< 4 =

hr ?x y 9:x 6= + ;y ; + 9=< 4 = hr 99x + 7y + ?6 4 =

hr 99x 7y ?6 4 =

Ajnkj, taj rjqulrjo jquitlhn hf taj tincjnt ls 99x 7y ?6 4 =

Jxidpbj # 99 5 Flno taj jquitlhn hf tincjnts th taj klrkbj x 7 + y7 6x + ?y 97 4 = walka irj piribbjb th taj

blnj ?x + ;y + : 4 =

[hbutlhn 5 Clvjn klrkbj ls x7 + y7 6x + ?y 97 4 = ... ....(l)

ino clvjn blnj ls ?x + ;y + : 4 = .......(ll)

Kjntrj hf klrkbj (l) ls (;, 7) ino lts riolus ls :. Jquitlhn hf iny blnj

?x + ;y + 4 = piribbjb th taj blnj (ll) .......(lll)

Lf blnj (lll) ls tincjnt th klrkbj, (l) tajn

77 ;?

|`)7(;;.?|

4 : hr |6 + `| 4 7:

hr 6 + 4 7: ` 4 9

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"dinlsa`udirpayslks.ln" 3

Jxidpbj# 975 Flno taj jquitlhn hf taj nhrdib th taj klrkbj x 7 + y7 :x + 7y ?1 4 = it taj phlnt (:, 6).

[hbutlhn 5 [lnkj nhrdib ls blnj ehlnlnc kjntrj

9,

7

:ino (:, 6)

[bhpj 4:

9?

Ajnkj, taj jquitlhn hf taj nhrdib it (:, 6) lsy 6 4 (9?/:) (x :) 9?x :y ?= 4 =

[jbf priktlkj prhgbjd 5

(99) Flno taj jquitlhn hf taj nhrdib th taj klrkbj x7 + y7 7x ?y + ; 4 = it taj phlnt (7, ;).

Inswjr 5 (99) x y + 9 4 =

]ilr hf tincjnts frhd i phlnt_aj jquitlhn hf i pilr hf tincjnts oriwn frhd taj phlnt I (x

9,y

9) th taj klrkbj

x7 + y7 + 7cx + 7fy + k 4 = ls 5 [[9

4 _.

Sajrj [ x7 + y7 + 7cx + 7fy + k 0 [9 x

9 + y

9 + 7cx

9+ 7fy

9+ k

_ xx9

+ yy9

+ c(x + x9) + f(y + y

9) + k.

Jxidpbj # 9; Flno taj jquitlhn hf taj pilr hf tincjnts oriwn th taj klrkbj x7 + y7 7x + ?y 4 = frhd taj phlnt

(=, 9)

[hbutlhn 5 Clvjn klrkbj ls [ 4 x7 + y7 7x + ?y 4 = ... ....(l)

Bjt ] (=, 9)Fhr phlnt ], [

94 =7 + 97 7.= + ?.9 4 :

Kbjirby ] bljs hutsloj taj klrkbj

ino _ x . = + y . 9 (x + =) + 7 (y + 9)l.j. _ x +;y + 7.Nhw jquitlhn hf pilr hf tincjnts frhd ](=, 9) th klrkbj (9) ls [[

94 _7

hr : (x7 + y7 7x + ?y) 4 ( x + ;y + 7)7

hr :x7 + :y7 9=x + 7=y 4 x7 +

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"dinlsa`udirpayslks.ln" 1

[jbf priktlkj prhgbjds 5

(97) Flno taj ehlnt jquitlhn hf taj tincjnts tarhuca (3, 9) th taj klrkbj x7 + y7 4 7:.

Inswjr 5 (97) 97x7 97y7 + 3xy 93:x 7:y + 67: 4 =

Bjncta hf i tincjnt ino phwjr hf i phlnt_aj bjncta hf i tincjnt frhd in jxtjrnib phlnt (x

9,y

9) th taj klrkbj

[ x7 + y7 + 7cx + 7fy + k 4 = ls clvjn gy B 4 kyf7xc7yx 997

97

9 4 9[ .

I] 4 bjncta hf tincjnt

I]7 4 IO . IJ

[quirj hf bjncta hf taj tincjnt frhd taj phlnt I ls ibsh kibbjo taj phwjr hf phlnt w.r.t. i klrkbj.

]hwjr hf i phlnt w.r.t. i klrkbj rjdilns khnstint.

]hwjr hf i phlnt ] ls phsltlvj, njcitlvj hr zjrh ikkhrolnc is taj phlnt I ls hutsloj, lnsloj hr hn taj

klrkbj rjspjktlvjby.

Jxidpbj # 9? Flno taj bjncta hf taj tincjnt oriwn frhd taj phlnt (:, 9) th taj klrkbj x 7 + y7 + 6x ?y ; 4 =

[hbutlhn 5 Clvjn klrkbj ls x7 + y7 + 6x ?y ; 4 = .........(l)

Clvjn phlnt ls (:, 9). Bjt ] 4 (:, 9)

Nhw bjncta hf taj tincjnt frhd ](:, 9) th klrkbj (l) 4 ;9.?:.69: 77 4 3

[jbf priktlkj prhgbjds 5

(9;) Flno taj irji hf taj quiorlbitjrib fhrdjo gy i pilr hf tincjnts frhd taj phlnt (?, :) th taj klrkbj

x7 + y7 ?x 7y 99 4 = ino i pilr hf lts rioll.

(9?) Lf taj bjncta hf taj tincjnt frhd i phlnt (f, c) th taj klrkbj x7 + y7 4 ? gj fhur tldjs taj bjncta hf

taj tincjnt frhd lt th taj klrkbj x7 + y7 4 ?x, sahw tait 9:f7 + 9:c7 6?f + ? 4 =

Inswjr. (9;) 1 s q. unlts

Olrjkthr k lrkbj_aj bhkus hf taj phlnt hf lntjrsjktlhn hf twh pjrpjnolkubir tincjnts ls kibbjo taj olrjkthr klrkbj hf taj

clvjn klrkbj. _aj olrjkthr klrkbj hf i klrkbj ls taj khnkjntrlk klrkbj aivlnc riolus jquib th 7 tldjs taj

hrlclnib klrkbj.

]rhhf 5

IK 4 r khsjk ?: 4 7r

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"dinlsa`udirpayslks.ln"