The gondola lift

A tangent and its normal in an arbitrary point on an ellipse with eccentricity e. The focal points are also shown. What is the minimum value of blue : red in terms of e?

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The minimal blue : red is (1-e)/(1+e).

Solution by Stéphane Bernard.

This proof makes use of the fact that the normal bisects the angle between tangency point and focal points (or Focus-to-focus reflection property).

Solution by Lewis Corner.

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