Tangent at $\left(x_1,y_1\right)$

$x{x}_1-2y{y}_1-4=0$

This is parallel to $2x-y=0$

$\Rightarrow \frac{x_1}{2y_1}=2$

$\Rightarrow x_1=4y_1 ...\left(1\right)$
Point $\left(x_1,y_1\right)$ lie on hyperbola $\frac{x_1^2}{4}-\frac{y_1^2}{2}-1=0 ...\left(2\right)$

Using equation $\left(1\right)$ and $\left(2\right)$, $4{\left(y_1\right)}^2-\frac{{y_1}^2}{2}=1\Rightarrow {y_1}^2=\frac{2}{7} \& {x_1}^2=\frac{32}{7}$

We get $x_1^2+5y_1^2=6$