If the circle x ^ 2 + y ^ 2 - 2gx + 6y - 19c = 0, g, c ∈ ℝ passes...

Given circle ${x}^{2} + {y}^{2} - 2gx + 6y - 19c = 0$

Also given circle passes through \left(6, 1\right)$</p><p>So,$12g + 19c = 43 ⋯ \left(1\right)

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

So, $g + 6c = 8 \ldots \left(2\right)$

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 $= 2\sqrt{{g}^{2} - C} = 2\sqrt{{2}^{2} + 19} = 2\sqrt{23}$