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Math Help - [SOLVED] Hyperbola 3

  1. #1
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    [SOLVED] Hyperbola 3

    Hello.

    Prove that the locus of midpoints of parallel chords of xy = c^2 is a diameter.

    Is anyone able to give me some hints on how to do this question?
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  2. #2
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    Hello UltraGirl
    Quote Originally Posted by UltraGirl View Post
    Hello.

    Prove that the locus of midpoints of parallel chords of xy = c^2 is a diameter.

    Is anyone able to give me some hints on how to do this question?
    Consider the chord joining P\;\Big(cp,\frac cp\Big) to Q\;\Big(cq,\frac cq\Big).

    You can easily show that its gradient is -\frac1{pq}.

    A set of parallel chords will all have the same gradient, m, say. So:
    -\frac1{pq}=m, where m is a constant. Call this equation (1).
    The mid-point (x,y) of PQ satisfies:
    x = \tfrac12c(p+q)
    and
    y = \tfrac12(\frac cp+\frac cq\Big)
    =\tfrac12c\Big(\frac{p+q}{pq}\Big)
    Now, using equation (1), find an equation connecting x, y and m, and show that this, the locus of the mid-point of PQ, is a straight line through the origin.

    Can you complete it now?

    Grandad
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