CGPET 2021 Solved Paper

Family of circles which pass through origin and their centre lies on Y-axis has C(0,k) and radius =k
Then, equation of circle will be
(x−0)2+(y−k)2=k2

x2+y2+k2−2ky=k2
x2+y2−2ky=0 ......(i)
Differentiating eq.(i) w.r.t.x,
2x+2y

dy

dx

−2k

dy

dx

=0.......(ii)
From eq. (i) , k=

x2+y2

2y

. Put it in eq.(ii),
2x+[2y−2(

x2+y2

2y

)]

dy

dx

=0
⇒ [2y2−x2−y2]

dy

dx

+2xy=0
⇒ (y2−x2)

dy

dx

+2xy=0
⇒ (x2−y2)

dy

dx

−2xy=0