Form the differential equation of the family of curve represented by $x^2 + y^2 = 2ax$, $a$ being the parameter.

$2xy\frac{dy}{dx}=y^2-x^2$

The correct answer is Option (2) → $2xy\frac{dy}{dx}=y^2-x^2$

Given $x^2 + y^2 = 2ax$, $a$ being the parameter   ...(1)

Differentiating it w.r.t. x, we get

$2x + 2y\frac{dy}{dx}= 2a$.

Putting this value of $2a$ in (1), we get $x^2 + y^2 = \left(2x+2y\frac{dy}{dx}\right)x$

$⇒ 2xy\frac{dy}{dx}= y^2 - x^2$, which is the required differential equation of the given family of curve.