Consider a hyperbola H   :   x 2 - 2 y 2 = 4 . Let the tangent at a point P ( 4 , 6 ) meet the x…

$\frac{x^2}{4}-\frac{y^2}{2}=1$

$e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{\frac{3}{2}}$

$\therefore$ Focus $F(ae,0)\Rightarrow F(\sqrt{6},0)$

equation of tangent at $P$ to the hyperbola is
$2x-y\sqrt{6}=2$

tangent meet $x$-axis at $Q(1,0)$

$\&$ latus rectum $x=\sqrt{6}$ at $R\left(\sqrt{6},\frac{2}{\sqrt{6}}\left(\sqrt{6}-1\right)\right)$

$\therefore$ Area of ${\Delta }_{\mathrm{QFR}}=\frac{1}{2}\left(\sqrt{6}-1\right)·\frac{2}{\sqrt{6}}\left(\sqrt{6}-1\right)$

$=\frac{7}{\sqrt{6}}-2$ sq. units