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

$x^2+y^2=36\text{ , then which one of the following is true? }$

$\begin{array}{l}y=mx+c\text{ is tangent to hyperbola }\Rightarrow c^2=a^2{m}^2-b^2=100m^2-64\\ y=mx+c\text{ is tangent to circle }\Rightarrow c^2=r^2\left(1+m^2\right)=36\left(1+m^2\right)\\ \therefore 100m^2-64=36+36m^2\end{array}$

It gives 

$m^2=\frac{25}{16}\text{ and }{c}^2=\frac{369}{4}\Rightarrow 4c^2=369$