I'm trying to understand the proof for eccentricity; I can get about halfway through it. The problem I'm having is summarized below
If e<1, you get the equation of an ellipse of the form
The foci of an ellipse are at a distance c from the center, where
This shows that
It follows from equations 4 and 5 that the eccentricity is given by
If someone could help explain how is derived, or really what it means I'd be so grateful. I'm using Calculus Early Transcendentals 6th Ed. by Stewart (Ch. 10, Sect. 6).