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.
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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