Thread: Eccentricity, trouble w/ the proof

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



where

Eq. 4:




The foci of an ellipse are at a distance c from the center, where

Eq 5:


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).

The conclusion is that the eccentricity of an ellips is: , now:
If you write:
Calculating with this:

Substitution of and out of eq 4 and you get:


Now take the square root of .