A line parallel to the straight line 2 x - y = 0 is tangent to the hyperbola x 2 4 − y 2 2 = 1 at the…

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 $\left(2\right)$,  $4y_1^2 - \frac{y_1^2}{2} = 1 \Rightarrow y_1^2 = \frac{2}{7}$ and $x_1^2 = \frac{32}{7}$ We get $x_1^2 + 5y_1^2 = 6$