Question

Let the circles S ≡ x² + y^{2 −} 2cy − a² = 0 and S' ≡ x² + y² − 2bx + a² = 0 whose centres are A and B, respectively, intersect at points P and Q. Show that the points P, Q, A, B and origin are concyclic.

Answer 1

Let L ≡ S − S' ≡ bx – cy − a² = 0. The required circle equation is of the form

This circle passes through origin  − a² − λ a² = 0 λ = −1. The required circle is

x² + y² − 2cy − a² − (bx − cy − a²) = 0

 x² + y² − bx − cy = 0

which also passes through the centres (0, c) and (b, 0) of the circles S = 0 and S'  0, respectively.