# How was substitution chosen in ellipse equation derivation?

1. Aug 23, 2011

### symbolipoint

I have been spending a few days reviewing parts of College Algebra from College Algebra, by Aufmann, Barker, & Nation. I am following the discussion of the shape, features, and equation for an ellipse, and I understand the derivation well, EXCEPT that I do not know how the subsitution of b2 = a2 - c2, where a is half the length of the major axis, b is half the length of the minor axis, and c is the length from the origin to either focus. The standard form for the ellipse being:
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

I tried drawing my own pictures, graphs, to figure out how the substitution was chosen, but I made no progress, although I see that the substituion works by just trusting it. The page in the book for this is 311 and 312.

This is NOT homework. I earned my credit in Pre-Calculus in college about three decades ago.

(This is my third edit. Sometimes the TEX equation works, sometimes the TEX equation stays with all the tags showing. This is inconsistant.)

Last edited: Aug 23, 2011
2. Aug 23, 2011

### symbolipoint

3. Aug 24, 2011

### symbolipoint

Finally some good progress in understanding, essentially the same information as in the book, in this online article, http://en.wikipedia.org/wiki/Proofs_involving_the_ellipse [Broken]

The reason for the substitution makes more sense now. Distances from one focus to point on ellipse plus from other focus to same point on ellipse are always 2a; when x=0, y=b; then both distances from foci to (0, b) are equal. This makes right angle ..., b2+c2=a2.

Last edited by a moderator: May 5, 2017