How to show that the angle at intersections of concentric ellipses with a radial line is constant?

Question

I'm looking for a proof that the angle at intersections of concentric ellipses with a radial line remains constant as you move from one concentric ellipse to another.

For example, a series of concentric ellipses of different sizes are centered on a point of origin. The radial lines from the point of origin outwards that trace the semi-major and semi-minor axis of the concentric ellipses only form right-angles at each intersection with the concentric ellipses.

How may one show that the other radial lines forming non-perpendicular intersections with the concentric circles form a constant angle with respect to the same radial line?

Answer 1

First of all, I don'r think the problem as you have stated it is true.

The ellipses must be more than concentric; parameters in the ellipses must also be in proportion.

With that addition to the problem, I think it becomes fairly straight forward.

The derivative at any point on the ellipse with parameters a & b at point (x₁,kx₁) is

-b²/(k²a²) and if the parameters are in proportion, this fraction will be the same on any other ellipse.

If I haven't made the answer clear, please send me a message & I will try to clarify.