Is the midpoint of an elipse's "cord", through the center, the midpoint of the curve?

Consider an elipse centered at (0, 0). It intersects the x-axis at (a, 0) and (-a, 0), and it intersects the y-axis at (0, b) and (0, -b). Construct a line segment from point (a, 0) to point (0, b), and call it segment A. Now, mark off the point that is the midpoint of this segment, call it M. Finnally, draw a line segment starting at the origin (0, 0), the center of the elipse, which intersects A at M, and draw this segement until it intersects the elipse and call this point on the elipse P. Is it neccesarily true, or not neccesarily true, that the point P is the "midpoint" of the distance along the elipse from (0, b) to (a, 0)?