Length of Latus Rectum in Ellipses: A Geometric Proof

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The length of the latus rectum in an ellipse is determined by a line segment through a focus, perpendicular to the major axis, with endpoints on the ellipse. Each ellipse has two latus recta, and their length can be expressed as 2b^2/a, where 'b' is the semi-minor axis and 'a' is the semi-major axis. To derive this, one can draw the ellipse, identify the relationships between the axes, and apply trigonometric principles. Substituting x = ae helps find the corresponding y-coordinate, which allows for calculating the length of the latus rectum. Understanding these geometric relationships is crucial for proving the length of the latus rectum in ellipses.
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"A line segment through a focus with endpoints on the ellipse and perpendicular to the major axis is a latus rectum of the ellipse. Therefore, an ellipse has two latus recta. Show that the length of each latus rectum is 2b^2/a."

I've been stuck on this for a little while now. Can anyone point me in the right direction?
 
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Draw an ellipse, show the relationships of A and B, do a little trig...
 
Substitute x = ae find y coordinate 2y will be the length of rectum
 
Thank you :smile:
 

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