The hyperbola can be defined by the following parametric equations:

.

Then .

The equation of the tangent can be written using the model : .

Use this equation to get the coordinates of the x- and y-intercepts, that is, the coordinates of P and Q. Use these coordinates to get the midpoint of PQ.

You now have the parametric equations of the required locus. Convert them into a Cartesian equation if required.

Warning:My carelessness is a well documented fact. The above results may contain errors. You should work through the calculations very carefully for yourself.