If the circle $x^{2} + y^{2} - 2gx + 6y - 19c = 0,; g,c \in \mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x - 2cy = 8$, then the length of intercept made by the circle on $x$-axis is :

$2\sqrt{23}$