Foci of an Ellipse

Date: 06/08/2001 at 13:52:00
From: Scott Nugent
Subject: Elliptical foci

If the major and minor axis of an ellipse are given, how do I find the 
focus points?

Date: 06/08/2001 at 15:40:43
From: Doctor Floor
Subject: Re: Elliptical foci

Hi, Scott,

Thanks for writing.

The foci F1 and F2 of an ellipse lie on its major axis, at equal 
distances from the center M of the ellipse. Let P be a variable point 
on the ellipse; then the sum of distances d(P,F1)+d(P,F2) is constant.

Now let the major axis meet the ellipse in X1 and X2 and the minor 
axis in Y1 and Y2. Let a be the length of the major axis. In a figure:

           Y1
           |
           |
 X1--F1----M-----F2--X2
           |
           |
           Y2

Clearly 

  d(X1,F1)+d(X1,F2) = d(X1,F1)+d(X2,F1) = a

and thus d(Y1,F1) = d(Y2,F2) = a/2.

Knowing this it is not too difficult to find the position of F1 and F2 
(for instance we might use Pythagoras' theorem in triangle F1MY1 to 
compute the distance from M to F1).

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/   

Date: 06/08/2001 at 15:43:45
From: Doctor Peterson
Subject: Re: Elliptical foci

Hi, Scott.

The foci lie along the major axis, at distance c on either side of the 
center, where

    c^2 = a^2 - b^2

You can find this information (and an explanation) by searching our 
archives for the words "ellipse major minor focus":

  Analytic Geometry Formulas
  http://mathforum.org/dr.math/faq/formulas/faq.analygeom_2.html   
  (select Two Dimensions: ellipses)

  Ellipses: Pythagorean Relationship
  http://mathforum.org/library/drmath/view/54802.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 06/08/2001 at 15:51:57
From: Scott Nugent
Subject: Re: Elliptical foci

Thank you very much!