If the line y=m x+c is a common tangent to

General tangent to hyperbola in slope form is

$y=m x \pm \sqrt{100 m^{2}-64}$

and the general tangent to the circle in slope form is

$y=m x \pm 6 \sqrt{1+m^{2}}$

For common tangent,

$36\left(1+m^{2}\right)=100 m^{2}-64$

$\Rightarrow 100=64 m^{2} \Rightarrow m^{2}=\frac{100}{64}$

$\therefore c^{2}=36\left(1+\frac{100}{64}\right)=\frac{164 \times 36}{64}=\frac{369}{4}$

$\Rightarrow 4 c^{2}=369$