If $P$ and $Q$ be a pair of conjugate points with respect to a circle $S$. Prove that the circle on $PQ$ as diameter cuts the circle $S$ orthogonally.

Two circles are drawn through the points $(a, 5a)$ and $(4a, a)$ to touch the axis of $y$, where $a>0$. Prove that they intersect at an angle ${\tan }^{-1}\left(\frac{40}{9}\right)$.

Find the condition that the chord of contact of tangents from the point $\left(\alpha ,\beta \right)$ to the circle $x^2+y^2=a^2$ should subtend a right angle at the centre. Hence find the locus of $\left(\alpha ,\beta \right)$.