Let the point $P\left(x_1, y_1\right)$, then
$$
\frac{10}{\sqrt{x_1^2+y_1^2-12 x_1-16 y_1}}=\frac{1}{\sqrt{3}}
$$


$\Rightarrow \quad x_1^2+y_1^2-12 x_1-16 y_1=300$

Since, line $A B, 5 y=5 x+k$, is a chord of contact of circle with respect to point $P\left(x_1, y_1\right)$, so,
$$
\begin{aligned}
x x_1+y y_1-6\left(x+x_1\right)-8\left(y+y_1\right) & =0 \\
x\left(x_1-6\right)+y\left(y_1-8\right)-\left(6 x_1+8 y_1\right) & =0 \\
\therefore \quad \frac{x_1-6}{-5}=\frac{y_1-8}{5}=\frac{6 x_1+8 y_1}{K} & =\lambda(\text { let })
\end{aligned}
$$

From Eqs. (i) and (ii), we are getting
$$
\begin{array}{rlrl}
& 25 \lambda^2+25 \lambda^2 =400 \Rightarrow \lambda= \pm 2 \sqrt{2} \\
& \lambda=\frac{6 x_1+8 y_1}{k} \\
\Rightarrow & K=\frac{6 x_1+8 y_1}{\lambda} =\frac{36-30 \lambda+64+40 \lambda}{\lambda} \\
\Rightarrow & k =\frac{100}{\lambda}+10 \Rightarrow k=10 \pm 25 \sqrt{2} \\
\Rightarrow & k =5(2 \pm 5 \sqrt{2})
\end{array}
$$