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?