Any conic section symmetric about the origin satisfies the equation

Using this, prove that if the conic section is an ellipse then we have

This can be interpreted to say that the sum of the distances from a point on an ellipse to the foci is a constant.

*Proof.* **Incomplete.**

let X=(x,y) and N=(1,0). Then XN=x and |ex|<=a. From that we have |eXN-a|=|ex-a|=a-ex. If we plug -F there to get the distance from the other focal point, and sum the equalities, we get 2a.