If the line $y=mx+c$  is a common tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ and the circle $x^2+y^2=36$,

 then which one of the following is true?

If $y = mx + c$ is a tangent to $\frac{x^2}{{10}^2}-\frac{y^2}{8^2}=1$, then

$\therefore c^2=100m^2-64$                                   …(i)

y=mx+c is also tangent for $x^2+y^2=36$,then $c^2=36\left(1+m^2\right)\to \left(ii\right)$

From (i) and (ii), we obtain 

$100m^2-64=36\left(1+m^2\right)\Rightarrow 64m^2=100\Rightarrow m=\pm \frac{5}{4}$

$\therefore c^2=36\left(1+\frac{25}{16}\right)=\frac{369}{4}$           $\left[\text{ Putting }m=\pm \frac{5}{4}\text{ in (ii) }\right]$

$\Rightarrow 4c^2=369$