Let us suppose equation of hyperbola is
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$e=2$ 

$\because b^2=\left(e^2-1\right)a^2$

$\Rightarrow b^2=3a^2$
It is passing through $\left(\mathrm{4,6}\right)$

$\frac{16}{a^2}-\frac{36}{b^2}=1$

$\Rightarrow a^2=4,b^2=12$
So, equation of tangent is $T=0$

$\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1$

Tangent at point $(4,6)$ is
$\Rightarrow x-\frac{y}{2}=1$
$\Rightarrow 2x-y-2=0$