KöMaL 2013 A584

In 3-space, let $S$ be a non-degenerate conic section which is not a circle. Consider the apices of those right circular conical surfaces that contain $S$. ($a$) Show that these points lie on a conic section, uniquely determined by $S$. ($b$) Denote by $C$($S$) the conic section that contains the possible apices. Prove that $C$ ( $C$($S$) ) =$S$ for arbitrary $S$.