Let P be the point of intersection of the common tangents to the...

Let $m$ is slope of common tangent and an equation of tangent to hyperbola $\dfrac{{x}^{2}}{1} - \dfrac{{y}^{2}}{8} = 1$ is

$y = mx \pm \sqrt{1 ⋅ {m}^{2} - 8} . . . \left(1\right)$

equation of tangent to parabola ${y}^{2} = 12x$ is

$y = mx + \dfrac{3}{m} . . . \left(2\right)$

$∵ \left(1\right) \text{ and } \left(2\right)$ is same

$∴ \pm \sqrt{{m}^{2} - 8} = \dfrac{3}{m}$

$⇒ {m}^{4} - 8{m}^{2} - 9 = 0 ⇒ {m}^{2} = - 1$ (Reject)

$⇒ {m}^{4} - 8{m}^{2} - 9 = 0 ⇒ {m}^{2} = 9 ⇒ m = \pm 3$

$∵$ equation of common tangents is $y = 3x + 1 \text{ and } y = - 3x - 1$

$∴$ point of intersection of common tangents is $$R\left(- \dfrac{1}{3}, 0\right)$ & Foci of hyperbolas are \left(3,0\right) \text{ and } \left(- 3,0\right)$</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/05222085-c988-48e0-ae09-0ff8c8aa63dc-image4.jpeg" style="width: 300px; height: 116px;" /></p><p>$⇒ \dfrac{- 3\lambda + 3}{\lambda + 1} = - \dfrac{1}{3} ⇒ \lambda = \dfrac{5}{4}$</p><p>Hence, the required ratio is$5: 4$.