Thread: [SOLVED] Hyperbola Once More - Sorry &gt;&lt;

1. [SOLVED] Hyperbola Once More - Sorry &gt;&lt;

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.

2. 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?