Given circle is

\( \Large \left(x-13\right)^{2}+y^{2}=169-K \)

Therefore, Centre is \( \Large \left(13,\ 0\right)\ and\ radius = \sqrt{169-K} \)

From \( \Large \triangle OCT,\ OC^{2}=OT^{2}+CT^{2} \)

Therefore, \( \Large 13^{2}=5^{2}+169-K \)

or K = 25