Family of circles is $\left(x-3\right)\left(x-6\right)+\left(y-7\right)\left(y-5\right)+\lambda$
$\left(2x+3y-27\right)=0$
i.e., $x^2+y^2+x\left(2\lambda -9\right)+y\left(3\lambda -12\right)+\left(53-27\lambda \right)=0$
Common chord of family of circles and the given circle is
$\left(-5x-6y+56\right)+\lambda \left(2x+3y-27\right)=0$
which represents family of line passing through point of intersection of the lines
$-5x-6y+56=0$ and $2x+3y-27=0$
The point of intersection is $\left(2,\mathrm{}\hspace{0.17em}\frac{23}{3}\right)$
$a=2$
$b=\frac{23}{3}$
$a+3b=2+23=25$