Explanation

(c)
The equation of normal at point $P\left( a\cos\theta,\ b\sin\theta \right)$ is
$ax\sec\theta - by\ coese\theta = a^{2} - b^{2}$
The point of intersection with coordinate axes are
$R\left( \frac{a^{2} - b^{2}}{a}\cos\theta,\ 0 \right)$ and $\left( 0,\ \frac{a^{2} - b^{2}}{a}\sin\theta \right)$
Now, $RP^{2} = \left\lbrack a\cos\theta - \left( \frac{a^{2} - b^{2}}{a} \right)\cos\theta \right\rbrack^{2} + b^{2}\sin^{2}\theta$
$= \frac{b^{2}}{a^{2}}(b^{2}\cos^{2}\theta + a^{2}\sin^{2}\theta)$
And ${RS}^{2} = \frac{a^{2}}{b^{2}}(b^{2}\cos^{2}\theta + a^{2}\sin^{2}\theta)$
$\therefore{RP}^{2}:{RS}^{2} = b^{4}:a^{4}$
$\Rightarrow RP:PS = b^{2}:a^{2}$