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Thread: Hyperbola Proof

  1. #1
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    Hyperbola Proof

    Here's the question.

    If P is any point on the hyperbola and PR and PS are drawn parallel to the asymptotes, the dPR x dPS = constant. Show this is true for the hyperbola:

    x^2/4 - y^2/4 = 1

    by choosing three different points for P. Can you prove the general case?

    P.S. I'm assuming that dPR means the distance from P to R.

    Here's the diagram

    Hyperbola Proof-ss-2013-11-04-11.50.58-.png

    Thanks in advance.
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  2. #2
    MHF Contributor
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    Re: Hyperbola Proof

    You could make the substitution

    x=\frac{\sqrt{2}}{2}(x'+y')
    y=\frac{\sqrt{2}}{2}(-x'+y')

    which is equivalent to considering new axes Ox'y' turned 45 degrees clockwise. In the new coordinate system, the equation will be x'y' = 2 and PR and PS will simply be coordinates of P. You should also show that the transformation from (x, y) to (x', y') preserves distances, which is easy.
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