CGPET 2013 Solved Paper

The equation of family of circles touching the y-axis at the origin is,
‌(x±h)2+y2=h2
⇒‌‌x2+y2+2xh=0 . . . (i)
On differentiating w.r.t. x, we get
2x+2y‌

dy

dx

+2h=0
or 2x+2y⋅y′+2h=0(∵−1‌

dy

dx

=y′)
⇒‌‌2h=−(2x+2yy′)
On putting the value of 2h in Eq. (i), we get
‌x2+y2−x(2x+2yy′)‌=0
⇒‌‌x2+y2−2x2−2xyy′‌=0
⇒‌‌2xyy′+x2‌=y2
which is the required differential equation.