For simplicity, let's say that the ellipse is horizontal, centered at with the following foci: E at and F at . Thus, the distance from each focus to the center is c.
The distance from a general point on the ellipse to E is given by .
The distance from P to F is given by .
Looking at the case in which P is a vertex of the ellipse and adding up the distances from this vertex to each focus, we see that the sum of these distances is .
Thus, we know that:
Considering the case when P is the co-vertex located at , we see that , so . Substituting , we get:
This is the standard equation for an ellipse centered at with horizontal semi-major axis a and semi-minor axis b.