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Is There A Generalization Of Brouwer's Fixed Point Theorem?

Ellipse: Find the fixed point?

  • If 'P' be a moving point on the ellipse x^2/25 + y^2/16 =1 in such a way that the tangent at 'P' intersect x = 25/3 at Q then circle on PQ as diameter passes through a fixed point.Find the fixed point.

  • Answer:

    It is a known fact that if a point P on an ellipse (or hyperbola or parabola) has a tangent meeting a directrix at Q, then QF and PF are orthogonal, where F is the focus associated to the directrix. Your question is a special case of this since the foci are (-3,0)(3,0) and the directrices are x = -25/3 and x = 25/3. You can look up pages 68 / 70. http://lm250.fr/Lm3232009-10polyref.pdf

Hearty Wallows at Yahoo! Answers Visit the source

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At (3,0). Will get back to this later. Very interesting geometrical fact. This is one of the best problems I've come across here on Y!A. Wow. Edit: I'll later post the messy "proof", but there has to be a much simpler way to prove this. This is fascinating.

Scythian1950

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