$C_1: x^2+y^2-6 x-6 y+13=0$
$C_2: x^2+y^2-8 y+9=0$
Equation of common chord $A B \equiv C_1-C_2=0$
$\begin{aligned}
& \therefore-6 x-6 y+13+8 y-9=0 \\
& \Rightarrow 6 x-2 y=4 \\
& \Rightarrow 3 x-y=2
\end{aligned}$
$\therefore \quad$ Directrix is $3 x-y-2=0 ... (i)$
$m_D=3$
$\therefore \quad$ Slope of axis $=\frac{-1}{3}$
Equation of axis $\Rightarrow y=\frac{-1}{3} x$
$\Rightarrow x+3 y=0... (ii)$
$\therefore \quad$ Foot of perpendicular on directrix is intersection of directrix and axis.
$\therefore \quad$ From (i) and (ii),
$x=\frac{3}{5}, y=\frac{-1}{5}$


$\begin{aligned} & \therefore \quad \frac{\alpha+\frac{3}{5}}{2}=0 \Rightarrow \alpha=\frac{-3}{5} \\ & \Rightarrow \frac{\beta-\frac{1}{5}}{2}=0 \Rightarrow \beta=\frac{1}{5}\end{aligned}$
$\therefore \quad$ Focus is $\left(\frac{-3}{5}, \frac{1}{5}\right)$.