If a circle C passing through the point (4, 0) touches the circle x...

The equation of the tangent to the circle ${x}^{2} + {y}^{2} + 2gx + 2fy + c = 0$ at a point \left({x}_{1}, {y}_{1}\right)$is$x{x}_{1} + y{y}_{1} + g\left(x + {x}_{1}\right) + f\left(y + {y}_{1}\right) + c = 0 .

Thus,$the equation of the tangent to{x}^{2} + {y}^{2} + 4x - 6y - 12 = 0$at \left(1, - 1\right) is

$x\left(1\right)$ + y\left(- 1\right)$+ 2\left(x + 1\right)$ - 3\left(y - 1\right)$- 12 = 0$

$⇒ 3x - 4y - 7 = 0 .</p><p>Also, we know that the equation of a circle touching a circle${x}^{2} + {y}^{2} + 2gx + 2fy + c = 0$and line$lx + my + n = 0$at \left({x}_{1}, {y}_{1}\right)$ is ${\left(x - {x}_{1}\right)}^{2} + {\left(y - {y}_{1}\right)}^{2} + \lambda\left(lx + my + n\right) = 0$

Hence, the required circle touching the line $3x - 4y - 7 = 0$ at \left(1, - 1\right)$is</p><p>${\left(x - 1\right)}^{2} + {\left(y + 1\right)}^{2} + \lambda\left(3x - 4y - 7\right) = 0$</p><p>Given, this circle passes through \left(4, 0\right)$

$⇒ 9 + 1 + \lambda\left(12 - 7\right) = 0$

$⇒ \lambda = - 2$

Hence, the equation of the circle is ${\left(x - 1\right)}^{2} + {\left(y + 1\right)}^{2} - 2\left(3x - 4y - 7\right) = 0$

$⇒ {x}^{2} + {y}^{2} - 8x + 10y + 16 = 0$

Also, we know that the radius of the circle ${x}^{2} + {y}^{2} + 2gx + 2fy + c = 0$ is $\sqrt{{g}^{2} + {f}^{2} - c}$

Hence, radius $= \sqrt{{\left(- 4\right)}^{2} + {5}^{2} - 16} = \sqrt{16 + 25 - 16} = 5$ units.