If the line y = mx + c is a common 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 ${c}^{2} = {a}^{2}{m}^{2} - {b}^{2}$ and the line is tangent to the circle ${x}^{2} + {y}^{2} = {r}^{2}$ is ${c}^{2} = {r}^{2}\left(1 + {m}^{2}\right) .$

Hence, for the given line and the hyperbola we have, ${c}^{2} = 100{m}^{2} - 64 . . . \left(i\right)$ and for the line and the circle, we have

${c}^{2} = 36\left(1 + {m}^{2}\right) . . . \left(ii\right)$

On comparing the above two equations, we get

$100{m}^{2} - 64 = 36 + 36{m}^{2}$

$64{m}^{2} = 100$

${m}^{2} = \dfrac{100}{64}$ $⇒ m = \dfrac{10}{8} = \dfrac{5}{4} .$

$⇒ {c}^{2} = 36\left(1 + \dfrac{25}{16}\right) = \dfrac{369}{4} .$