A Question on Ellipses, Part 1

It is well known that the tangent lines to a circle are those lines that intersect the circle only once. The normal lines are perpendicular to the tangent lines at the point of tangency. All normal lines of a circle pass through the center.

If we stretch the circle into an ellipse, the center divides into two foci. The ellipse still has tangent lines and normal lines, but there is no longer enough symmetry to expect that all normal lines will pass through the same point.

So where do the normal lines intersect? To make the question well-defined, we need to indicate what it is they are intersecting. The natural choice is the major axis, the line connecting the foci. (The major axis is horizontal in the picture above.) But the question is not interesting yet because the major axis itself is a normal line to the ellipse (at which points?), making the answer “the entire major axis”. So, the revised question is:

What is the set of points of intersection of a normal line of an ellipse with its major axis, barring the normal lines at the endpoints of the major axis?

Before you do any calculation, make an intuitive guess. I’ve given this problem to several mathematicians, and so far no one’s instincts were correct.