Question
Form the differential equation of the family of circles touching the y-axis at origin.
Solution
The equation of family of circles touching y-axis at origin is
(x – h)2 + y2 = h2 where h is arbitrary constant ...(1)
[∵ if (h, 0) is centre of any member of family, then its radius = h]
Differentiating (1) w.r.t.x, 
⇒ x – h + y y1= 0 ⇒ h = x + y y1
Putting value of h in (1), we get,
(x – x – y y1)2 + y2 = (x + y y1)2 ⇒ y2 y12 + y2 = y2 + 2 x y y1 + y2 y12
⇒ x2 – y2 + 2 x y y1 = 0 , which is the required differential equation.
