91Ó°ÊÓ

If a variable chord of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is a tangent to the circle \(x^{2}+y^{2}=c^{2}\) then prove that the locus of its middle point is \(\left(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)^{2}=c^{2}\left(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}\right) .\)

Obtain the equation of the chord joining the points \(\theta\) and \(\phi\) on the hyperbola in the form \(\frac{x}{a} \cos \left(\frac{\theta-\phi}{2}\right)-\frac{y}{b} \sin \left(\frac{\theta+\phi}{2}\right)=\cos \left(\frac{\theta+\phi}{2}\right)\). If \(\theta-\phi\) is a constant and equal to \(2 \alpha\), show that \(P Q\) touches the hyperbola \(\frac{x^{2} \cos ^{2} \alpha}{a^{2}}-\frac{y^{2}}{b^{2}}=1\).