A Perpendicularity on an Ellipse

Let $O$ be the center of an ellipse, and $A$ a point on the ellipse other than a vertex. Drop the perpendicular from $A$ to the major axis, with foot $M$. Extend $AO$ to meet the ellipse again at $B$, then extend $BM$ to meet the ellipse at a point $C$. Does there exist an ellipse for which $BA\perp CA$? Justify your answer.