From a point P, two tangents PA and PB are drawn to the hyperbola...

Let, $P$ be \left(h, k\right)$and the equation of tangent through$P$on the hyperbola is<br />$y = mx \pm \sqrt{a^{2}m^{2} - b^{2}}$<br />$⇒ { \left(k^{2} - mh\right)}^{2} =\left(a^{2}m^{2} - b^{2}\right)
$⇒ \left(h^{2} - a^{2}\right)m^{2} - 2khm + k^{2} + b^{2} = 0$ where roots are ${m}_{1} and {m}_{2}$

$⇒ \dfrac{k^{2} + b^{2}}{h^{2} - a^{2}} =$1 = {m}_{1}{m}_{2}$<br />$⇒$locus is$x^{2} - y^{2} = a^{2} + b^{2}$<br />$⇒$eccentricity$= \sqrt{1 + 1} = \sqrt{2}$</p>$