I am struggeling with the following problem:
The tangent at the point P(ct, c/t) (t>0) on the hyperbola meets the x-axis at A and the y-axis at B. The normal at P to the rectangular hyperbola meets the line y = x at C and the line y = -x at D.
The normal at P meets the hyperbola again at Q and the mid point of PQ is M. Prove that as t varies, the point M lies on the curve
I have calculated the points as follows:
I have the point Q at and the mid point of PQ, point M at
The locus of M is usually found by eliminating T from the coordinates of the point, but I can't seem to get anywhere. I end up with and we have but I can't eliminate t completely. Should I try working back from the given result? I feel there has to be a better way!