A line parallel to the straight line 2x - y = 0 is tangent to the...

Tangent at $\left({x}_{1}, {y}_{1}\right)$

$x{x}_{1} - 2y{y}_{1} - 4 = 0$

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

$⇒ \dfrac{{x}_{1}}{2{y}_{1}} = 2$

$⇒ {x}_{1} = 4{y}_{1} . . . \left(1\right)<br />Point$\left({x}_{1}, {y}_{1}\right) lie on hyperbola ${x}_{1}^{2}4 - {y}_{1}^{2}2 - 1 = 0 . . . \left(2\right)$

Using equation \left(1\right)$and \left(2\right)$, $4{\left({y}_{1}\right)}^{2} - \dfrac{{{y}_{1}}^{2}}{2} = 1 ⇒ {{y}_{1}}^{2} = \dfrac{2}{7} \text{\text{ and } } {{x}_{1}}^{2} = \dfrac{32}{7}$

We get ${x}_{1}^{2} + 5{y}_{1}^{2} = 6$