Sol. Let y = f(x) be the curve and let P(x, y) be any point on it. Then, the length of the subnormal at P = y .

It is given that Length of the subnormal at P ∝ OP 2 ⇒ y = λ OP

2 , λ is the constant of proportionality

⇒ y = λ (x 2 + y 2 )

⇒ 2y –2 λ y 2 = 2 λ x 2 –2 λ y

2 = 2 λ x 2 ⇒ –2 λ v = 2 λ x

2 .. (i)

where v = y 2 This is a linear differential equation I.F. = =

Multiplying (i) by and integrating, we obtain

= dx

⇒ = 2 λ

⇒ ve

–2 λ x = 2 λ

+ C

⇒ ve –2 λ x = –x 2 e –2 λ x – e –2 λ x – e –2 λ x +C

⇒ y 2 = –x 2 – – + C

This is the required family of curves.