# Geo5.3.46

From Example Problems

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

Then

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.