Focus reflections in a conic?

Question

Point F is a focus of an ellipse. Point P is the image of F when it is reflected on a line that is tangent to the ellipse. What is the locus of P as F is reflected on all tangent lines? Give a detailed description please.

Answer the same question again for the cases of a parabola and a hyperbola.

Answer 1

Favourite answer

Let's denote the other focus E and the point of tangency A

Since tangent line forms equal angles with lines FA and EA, all three points E,A and P lie on on one straight line. Which means that

EP = EA + AP = EA + FA

On the other hand definition of ellipse is

EA + FA = const = major axis

Therefore all points P satisfy the condition

EP = const = R = major axis

that is the answer is

A circle centered at the other focus of radius R equal to major axis of ellipse

The same is true for hyperbolas, but in case if a parobola the answer is straight line parallel to directrix, which is a circle of infinite radius so to speak anyway