Ellipses: Pythagorean Relationship

Date: 2/12/96 at 14:0:31
From: Anonymous
Subject: Ellipses: Pythagorean Relationship

In an ellipse with major axis of 2a, minor axis of 2b, and foci c (on 
the major axis), the relationship c squared = a squared - b squared 
holds true (often used for finding the foci).  However, on further 
examination I was unable to visually understand how the three numbers 
fit into a Pythagorean relationship.  C is not a hypotenuse, lying on 
the same line as a.

I have consulted my math teacher, and looked in numerous books for an
illustrated explanation, but in vain. 

Thanks so much,

Brett

Date: 9/13/96 at 18:10:8
From: Doctor Ceeks
Subject: Re: Ellipses: Pythagorean Relationship

Hi,

Draw the two lines which connect the foci to one of the points
where the minor axis intersects the ellipse.

You will see two congruent right triangles, back to back.  It's
easy to see that the short sides of these right triangles are b and c.
I claim that the hypotenuse is a.

The reason is that twice that length is the distance from one focus,
to the ellipse, to the other focus, and it is a defining property
of an ellipse that such lengths are constant. So this distance is the
same as going from one focus to a point where the major axis 
intersects the ellipse, to the other focus... and that distance, you 
can see, is 2a.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/