All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
Simple properties of reflections in ellipse. (Posted on 2010-05-05) Difficulty: 5 of 5
Given an ellipse defined by its foci and major axis.

Prove that any line inside the ellipse not passing through the line segment joining the two foci when reflected off the ellipse would be tangent to the same inner ellipse as the initial line, with the same pair of foci.

Accordingly, if the line were to originally pass through the line segment joining the foci, the reflection on the ellipse would be tangent to the same hyperbola as the initial line, with the same pair of foci as the original ellipse.

The other possibility would be if the line started of from one focus, its reflection then passes through the other focus. (This degenerate case is known as the optical property of the ellipse, whose proof is much simpler.)

See The Solution Submitted by Vee-Liem Veefessional    
Rating: 4.6667 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Fifth of May, 2010. Comment 5 of 5 |
(In reply to Fifth of May, 2010. by Vee-Liem Veefessional)

I used the analytic approach because I am not familiar with the synthetic approach when working with conics. For example,

Given an ellipse and a point outside it, construct the lines through the point and tangent to the ellipse.

Given an ellipse and a point on it, construct the tangent line to the ellipse at that point.

The same two for a hyperbola.


  Posted by Bractals on 2010-05-06 16:42:22
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information