Coordinate Geometry: JEE Main 2022 Mathematics question

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