An ellipse with focal points further and further...

Question

Let's start with a definition of an ellipse that states: An ellipse is a set of points where each point's sum of distances from two focal points is equal to a constant value.

Now, we have got a two focal points and a given constant that creates an ellipse. Then we make those focal points further and further away. In my understanding the ellipse will become more and more "flat".

I have got two questions:

• How an eccentricity of the ellipse changes?
• If we reach a point where focal points are separated with a distance equal to the constant, does it mean our ellipse "degenerates" into a line? Or maybe a point?

Answer 1

Eccentricity is a measure of how much an ellipse has been "squished" from a circle. A circle has not been squished from itself at all, so its eccentricity is 0. A circle that has been completely squished becomes a line, which has an eccentricity of 1. You can play around with it here: https://www.mathopenref.com/ellipseeccentricity.html The answer to your second question is yes (it degenerates into a line).

Answer 2

I remember that for a conic with a focus at the origin, if the directrix is $x=\pm p$ where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation

$$r=\frac{ep}{1\pm e \cos \theta}$$ and when $e=1$ we not have a stright line but a parabola.