VMO 1964 Problem 3

Let $P$ be a plane and two points $A \in (P),O \notin (P)$ . For each line in $(P)$ through $A$ , let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$ . Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$ . Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$ , intersect $(C)$ by circles. Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$ .