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Thread: [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 $\displaystyle P\;\Big(cp,\frac cp\Big)$ to $\displaystyle Q\;\Big(cq,\frac cq\Big)$.

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

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

    Can you complete it now?

    Grandad
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