Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$

These ellipses take on the form

$$\sqrt{(x+1)^2 + y^2} + \sqrt{(x-1)^2+y^2}= K $$

where $2 < K < \infty$ (the case of $K = 2$ is a degenerate case).

You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).

Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)

What are the "normal curves" of the collection of ellipses i've listed?


One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.


It may not be mathematically rigorous, but look at the reflection laws of conic sections.

In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.

We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.


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