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?