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

Let the equation of tangents are
$y = {m}_{1}x + {C}_{1},$ and
$y = {m}_{2}x + {C}_{2}$
which cuts the coordinate axes at $E, F, G, H$ as shown in the figure
Now, $OE \times OG = OF \times OH$
$⇒$\left(\dfrac{- {C}_{1}}{{m}_{1}}\right) \times $\left(\dfrac{+ {C}_{2}}{{m}_{2}}\right) = \left({C}_{1}\right)\left(- {C}_{2}\right)⇒ {m}_{1}{m}_{2} = 1$
Let $P$ be \left(h, k\right)$and equation of the tangent through$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)
$⇒ \left(h^{2} - a^{2}\right)m^{2} - 2khm + k^{2} + b^{2} = 0$
whose roots are {m}_{1} and ${m}_{2} ⇒ {m}_{1}{m}_{2} = \dfrac{k^{2} + b^{2}}{h^{2} - a^{2}} =$1$<br />$⇒$locus is$x^{2} - y^{2} = a^{2} + b^{2}