If a tangent at a point P to a hyperbola meets the asymptotes in Q and R show that P is the midpoint of QR.
Let P be a point on the hyperbola
Equation to the pair of asymptotes of the hyperbola is
Treating a pair of lines as a degenerate conic,the equation of its chord having P as its midpoint is
This represents the equation of the tangent at P to the hyperbola.
Thus the length of tangent to a hyperbola intercepted between asymptotes is bisected at its point of contact.