Let a line L: 2x + y = k, k > 0 be a tangent to the hyperbola x ^ 2...

A line $y = mx + c$ is a tangent to the hyperbola $\dfrac{{x}^{2}}{{a}^{2}} - \dfrac{{y}^{2}}{{b}^{2}} = 1$ if ${c}^{2} = {a}^{2}{m}^{2} - {b}^{2} .$

Given that, tangent to hyperbola $\dfrac{{x}^{2}}{3} - \dfrac{{y}^{2}}{3} = 1$ is $2x + y = k$ or $y = - 2x + k$

Thus, we have, slope $m = - 2, c = k \text{ and } {a}^{2} = {b}^{2} = 3$

${k}^{2} = 3{\left(- 2\right)}^{2} - 3$

$⇒ {k}^{2} = 9$

Given $k > 0, ⇒ k = 3 .$

Thus, the equation of the tangent to the hyperbola is $y = - 2x + 3 .$

Given this line is also a tangent to the parabola, ${y}^{2} = \alphax$ and a line $y = mx + c$ is tangent to the parabola ${y}^{2} = 4Ax,$ if $c = \dfrac{A}{m}$

Thus, we have $3 = \dfrac{\left(\dfrac{\alpha}{4}\right)}{- 2}$

$⇒ \dfrac{\alpha}{- 8} = 3$

$⇒ \alpha = - 24 .$