Thank you for reporting, we will resolve it shortly
Q.
If the circle $x^2+y^2-2 g x+6 y-19 c=0, g$, $c \in R$ 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
Given circle $x^2+y^2-2 g x+6 y-19 c=0$
Passes through $(6,1)$
$12 g +19 c =43.....$(1)
Centre $( g ,-3)$ lies on given line
So, $g+6 c=8 .....$(2)
Solve equation (1) \& (2)
$c =1 \& g =2$
equation of circle $x^2+y^2-4 x+6 y-19=0$
Length of intercept on $x$-axis
$=2 \sqrt{g^2-c}=2 \sqrt{23}$