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Math Help - [SOLVED] Hyperbola Once More - Sorry ><

  1. #1
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    [SOLVED] Hyperbola Once More - Sorry ><

    PN is the perpepndicular to an asymptote from a point on a rectangular hyperbola. Prove that the locus of the midpoint of PN is a rectangular hyperbola with the same asymptotes.

    I've been staring at the diagram I've drawn for aaaaaaaaaaaaaaages but I can't think of a way to do this. Any hints appreciated.
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  2. #2
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    If you use an hyperbola with x-axis and y-axis as asymptotes, such as y = 1/x, your points look like this:

    Point on the Parabola: (s,t) = (s,1/s)

    Perpendicular to horizontal asymptote intersects at: (s,0)

    Perpendicular to vertical asymptote intersects at: (0,1/s)

    What do your midpoints look like?

    Have we lost anything by using the simplest hyperbola?
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