Doubt from Ellipse

Let R(h,k) be the second point of intersection of the two circles.

The idea here is that if M_1 is the midpoint of OP which is the centre of the circle with OP as diameter, and similarly M_2 that of OQ then M_1M_2 is the perp bisector of OR and hence passes through \left(\dfrac{h}{2}, \dfrac{k}{2} \right)

\therefore equation of M_1M_2 is hx+ky=\dfrac{h^2+k^2}{2}

Now if we have P(a\cos \theta, b\sin \theta) and Q(-a\sin \theta, b \cos \theta) then M_1\left(\dfrac{a\cos \theta}{2}, \dfrac{b \sin \theta}{2} \right) and M_2 \left(\dfrac{-a\sin \theta}{2}, \dfrac{b \cos \theta}{2} \right) lie on the above line

So ha \cos \theta+k b \sin \theta =h^2+k^2 and
-ha \sin \theta+ k b \cos \theta = h^2+k^2

Squaring and adding gives a^2h^2+b^2k^2 = 2(h^2+k^2)^2

Hence R lies on 2(x^2+y^2)^2 = a^2x^2+b^2 y^2