Putnam 1941 B4

Given two perpendicular diameters $AB$ and $CD$ of an ellipse, we say that the diameter $A'B'$ is conjugate to $AB$ if $A'B'$ is parallel to the tangent to the ellipse at $A$ . Let $A'B'$ be conjugate to $AB$ and $C'D'$ be conjugate to $CD$ . Prove that the rectangular hyperbola through $A', B', C'$ and $D'$ passes through the foci of the ellipse.