[Tony]

$8.$
$By \ \ the \ \ Law\ \ of \ \ Cosine$
$c^{2}=a^{2}+b^{2}-2ab\cos C$
$24c^{2}=ab\cos C$

$By \ \ the \ \ Law\ \ of \ \ Sine$
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

$=\frac{\cot C}{\cot A+\cot B}$

$=\frac{\frac{\cos C}{\sin C}}{\frac{\cos A}{\sin A}+\frac{\cos B}{\sin B}}$

$=\frac{\frac{\cos C}{\sin C}}{\frac{\sin C}{\sin A\sin B}}$

$= \frac{\cos C \sin A \sin B}{\sin^{2} C}$

$= 24$