Explanation

(a)
Given, $y^{2} = ax^{2} + bx + c \Rightarrow 2y\frac{dy}{dx} = 2ax + b$
$\ \Rightarrow 2\left( \frac{dy}{dx} \right)^{2} + 2y\left( \frac{d^{2}y}{dx^{2}} \right) = 2a\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$\Rightarrow y\frac{d^{2}y}{dx^{2}} = a - \left( \frac{dy}{dx} \right)^{2}$
$\Rightarrow y\frac{d^{2}y}{dx^{2}} = a - \left( \frac{2ax + b}{2y} \right)^{2}$
$\Rightarrow \ y\frac{d^{2}y}{dx^{2}} = \frac{4ay^{2} - (2ax + b)^{2}}{4y^{2}}$
$\Rightarrow 4y^{3}\frac{d^{2}y}{dx^{2}} = 4a\left( ax^{2} + bx + c \right) - (4a^{2}x^{2} + 4abx + b^{2})$
$\Rightarrow 4y^{3}\frac{d^{2}y}{dx^{2}} = 4ac - b^{2}$
$\Rightarrow y^{3}\frac{d^{2}y}{dx^{2}} = \frac{4ac - b^{2}}{4} =$constant