# Geo5.3.46

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 ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1\,$

Then ${\frac {x_{1}^{{2}}}{a^{2}}}-{\frac {y_{1}^{{2}}}{b^{2}}}=1\,$

Equation to the pair of asymptotes of the hyperbola is

$S\equiv 0\,$

Treating a pair of lines as a degenerate conic,the equation of its chord having P as its midpoint is $S_{1}=S_{{11}},{\frac {xx_{1}}{a^{2}}}-{\frac {yy_{1}}{b^{2}}}={\frac {x_{1}^{{2}}}{a^{2}}}-{\frac {y_{1}^{{2}}}{b^{2}}}\,$

${\frac {xx_{1}}{a^{2}}}-{\frac {yy_{1}}{b^{2}}}=1\,$

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.