Geo5.3.46

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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.

Main Page:Geometry:Hyperbola