SHARYGIN 2006 Problem 24

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$ . Find the geometric locus of the projections of $P$ on the lines $AB$ . b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$ . Find the geometrical locus of the projections $P$ on the $ABC$ plane