If the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{gx}+6\mathrm{y}-19\mathrm{c}=0, \mathrm{g}, \mathrm{c}\in \mathrm{ℝ}$ 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

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

Passes through (6, 1)

12g + 19 c = 43          ….(1)

Centre (g, -3) lies on given line

So, g + 6c = 8              ....(2)

Solve equation (1) & (2)

c = 1 & g = 2

equation of circle ${\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}+6\mathrm{y}-19=0$

Length of intercept on x-axis

$=2\sqrt{{\mathrm{g}}^2-\mathrm{c}}=2\sqrt{23}$