Ellipses: Pythagorean RelationshipDate: 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/ |
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