The condition that the chord \( \Large x \cos \alpha +y \sin \alpha - p = 0 \) of \( \Large x^{2}+y^{2}-a^{2}=0 \) may subtend a right angle at the centre of circle is:

The combined equation of the lines joining the origin to the points of intersection of \( \Large x \cos \alpha + y \sin \alpha = p\ and\ x^{2}+y^{2}-a^{2}=0 \) is a homogenous equation of second degree given by

\( \Large x^{2}+y^{2}-a^{2} \left(\frac{x \cos \alpha + \sin \alpha }{p}\right)^{2} = 0 \)

=> \( \Large \left[ x^{2} \left(p^{2}-a^{2}\cos^{2} \alpha \right) \right] \) + \( \Large \left(p^{2}-a^{2} \sin^{2} \alpha \right) - 2xya^{2} \sin \alpha \cos \alpha = 0  \)

The lines given by this equation are at right angle if

\( \Large \left(p^{2}-a^{2}\cos^{2} \alpha \right) + \left(p^{2}-a^{2}\sin^{2} \alpha \right) = 0 \)

=> \( \Large 2p^{2} = a^{2} \left(\sin^{2} \alpha +\cos^{2} \alpha \right) \)

=> \( \Large a^{2}= 2p^{2} \)