Diameter of an ellipse

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# Diameter of an ellipse

Submitted by edwardsj on Tue, 03/04/2014 - 21:32

Forums:

I am searching for a source demonstrating that, for any set of parallel^{} chords spanning an ellipse^{}, the longest chord passes through the center of the ellipse. I am not referring to the major and minor axes, which I know are the longest and shortest diameters^{}. Rather, I am referring to any set of parallel chords and want to show that the longest chord is a diameter that passes through the center. Any sources would be much appreciated!

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## diameter of ellipse

Would you have some hint in http://planetmath.org/conjugatediametersofellipse ? Unfortunately, the picture is corrupt.

## Thanks for your suggestion.

Thanks for your suggestion. I have studied the page you recommended, but it does not explicitly address what I want to establish.

## Hi Edwards, I don't know if

Hi Edwards, I don't know if this is still actual, but here is a simple way to prove it.

Start writing down the (square of) the distance of two any points in the plane, as a function of their 4 coordinates. There are four constraints on these points. They both have to be on a given ellipse and they both have to be on a straight line of given inclination m. Now use Lagrange multipliers to maximize the distance as a function of the coordinates and the position of the line (given, for example, by its crossing point with the x axis). The rest is straightforward.