Explanation

(d)
Since$\ \sin{\theta,\cos\theta}$ are the roots of $ax^{2} - bx + c = 0$
$\therefore\sin{\theta + \cos{\theta = \frac{b}{a}}}$
and$\ \sin{\theta\cos{\theta = \frac{c}{a}}}$
$\Rightarrow \sin^{2}{\theta + \cos^{2}{\theta + 2\sin{\theta\cos\theta = \frac{b^{2}}{a^{2}}}}}$
and$\ \sin{\theta\cos{\theta = \frac{c}{a}}}$
$\Rightarrow 1 + 2\left( \frac{c}{a} \right) = \frac{b^{2}}{a^{2}}$
$\Rightarrow b^{2} - a^{2} = 2ac$