Let $r = a-c$, where $a$ is the length of semi major axis and $c$ is the distance between origin and one of the foci of an ellipse.
I'm wondering if there a nice way to see that the below circle is always interior to the ellipse. $$(x-c)^2 + y^2 = r^2$$
I think this is equivalent to saying that the minimum distance between a focus and a point on the ellipse is $a-c$. If so, is there a way to prove this using geometry or some other means ? I feel calculus is messy here... Thank you!