If the circle x 2 + y 2 - 2 g x + 6 y - 19 c = 0 ,   g , c ∈ ℝ passes through the point 6 ,…

Given circle $x^2+y^2-2gx+6y-19c=0$

Also given circle passes through $\left(6,1\right)$

So, $12g+19c=43 \cdots \left(1\right)$

And centre $\left(g,-3\right)$ lies on $x-2cy=8$,

So, $g+6c=8 \dots \left(2\right)$

On solving equation $\left(1\right)$ and $\left(2\right)$ we get, $c=1$ and $g=2$

Now equation of circle will be,  $x^2+y^2-4x+6y-19=0$

Now length of intercept on $x$-axis is given by $=2\sqrt{g^2-C}=2\sqrt{2^2+19}=2\sqrt{23}$