A tangent line L is drawn at the point (2, - 4) on the parabola y ^...

Given, Parabola ${y}^{2} = 8x and circle${x}^{2} + {y}^{2} = {a}^{2}$</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/c207ecc1-93ee-40d3-a9cc-e37eb5553543-image.png" style="width: 257px; height: 214px;" /></p><p> Parabola${y}^{2} = 8x$at point \left(2, - 4\right)$

Differentiating w .r. t. x ' '

$2y\dfrac{dy}{dx} = 8$

$\dfrac{dy}{dx} = \dfrac{4}{y}$

$\dfrac{dy}{dx} \left(2, - 4\right) = - 1$

Equation of tangent

$y - {y}_{1} = \dfrac{dy}{dx}\left(x - {x}_{1}\right)$

$y + 4 = \left(- 1\right)\left(x - 2\right)$

$x + y + 2 = 0$

Line L is also tangent to the circle ${x}^{2} + {y}^{2} = {a}^{2}$

So, $OA = \sqrt{a}$ at \left(0, 0\right)$</p><p>\left|\dfrac{1 \times 0 + 1 \times 0 + 2}{\sqrt{2}}\right| = \sqrt{a}$

$\sqrt{2} = \sqrt{a}$

$⇒ a = 2$