BG-MO 1990 Problem 2

Let be given a real number $\alpha\ne0$ . Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$ , segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).