Let the eccentricity of the hyperbola H: x ^ 2 a ^ 2 - y ^ 2 b ^ 2...

Given equation of the hyperbola is $\dfrac{{x}^{2}}{{a}^{2}} - \dfrac{{y}^{2}}{{b}^{2}} = 1$

If $e$ is eccentricity of the hyperbola, then ${e}^{2} = 1 + \dfrac{{b}^{2}}{{a}^{2}}$

$1 + \dfrac{{b}^{2}}{{a}^{2}} = \dfrac{5}{2}$$⇒ {b}^{2} = \dfrac{3{a}^{2}}{2}$

Length of the latus-rectum is given by $\dfrac{2{b}^{2}}{a}$

i.e. $\dfrac{2{b}^{2}}{a} = 6\sqrt{2}$

$\dfrac{2}{a} \times \dfrac{3{a}^{2}}{2} = 6\sqrt{2}$

$⇒ a = 2\sqrt{2}$

and ${b}^{2} = \dfrac{3}{2} \times 8 = 12$

$⇒ b = 2\sqrt{3}$

If $y = mx + c$ is a tangent to the hyperbola, then ${c}^{2} = {a}^{2}{m}^{2} - {b}^{2}$

$∴ {c}^{2} = 4 \times 8 - 12$

Hence ${c}^{2} = 20$