ISL 1989 Problem 42

Let $A$ and $B$ be fixed distinct points on the $X$ axis, none of which coincides with the origin $O(0, 0),$ and let $C$ be a point on the $Y$ axis of an orthogonal Cartesian coordinate system. Let $g$ be a line through the origin $O(0, 0)$ and perpendicular to the line $AC.$ Find the locus of the point of intersection of the lines $g$ and $BC$ if $C$ varies along the $Y$ axis. Give an equation and a description of the locus.