# Normal Curves of Ellipses

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?

## Work:

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.