Explanation

(c)
Equation of family of ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$
$\Rightarrow$ $\frac{2x}{a^{2}} + \frac{2y}{b^{2}}.\frac{dy}{dx} = 0$
$\Rightarrow$ $\frac{x}{a^{2}} + \frac{y}{b^{2}}.\frac{dy}{dx} = 0$ …(i)
$\Rightarrow$ $\frac{1}{a^{2}} + \frac{y}{b^{2}}.\frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{2}\ \frac{1}{b^{2}} = 0$
$\Rightarrow$ $\frac{b^{2}}{a^{2}} + y\left( \frac{d^{2}y}{dx^{2}} \right) + \left( \frac{dy}{dx} \right)^{2} = 0$
$\Rightarrow$ $- \frac{y}{x}.\ \frac{dy}{dx} + y\ \frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{2} = 0$
[from Eq. (i), $\frac{b^{2}}{a^{2}} = - \frac{y}{x}.\frac{dy}{dx}$]
$\Rightarrow$ $xy\ \frac{d^{2}y}{dx^{2}} + \frac{dy}{dx}\left( x\ \frac{dy}{dx} - y \right) = 0$