Explanation

(a)
The combined equation of the lines joining the origin to the points of intersection of $x\cos\alpha + y\sin\alpha = p$ and $x^{2} + y^{2} - a^{2} = 0$ is a homogeneous equation of second degree given by
$x^{2} + y^{2} - a^{2}\left( \frac{a\cos\alpha + y\sin\alpha}{p} \right)^{2} = 0$
$\Rightarrow \lbrack x^{2}\left( p^{2} - a^{2}\cos^{2}\alpha \right) + y^{2}\left( p^{2} - a^{2}\sin^{2}\alpha \right) - 2xya^{2}\sin\alpha\cos\alpha = 0\rbrack$
The lines given by this equation are at right angle, if
$\left( p^{2} - a^{2}\cos^{2}\alpha \right) + \left( p^{2} - a^{2}\sin^{2}\alpha \right) = 0$
$\Rightarrow 2p^{2} = a^{2}(\sin^{2}\alpha + \cos^{2}\alpha)$
$\Rightarrow a^{2} = 2p^{2}$