If a circle C , whose radius is 3 , touches externally the circle x...

The equation of a tangent to the given circle at $\left(2, 2\right)$ is

$x\left(2\right) + y\left(2\right) + \left(x +2\right) - 2 \left(y +2\right) - 4=0⇒3 x - 6=0⇒ x =2$

Family of the circle touching a given line at a given point

${\left(x - 2\right)}^{2} + {\left(y - 2\right)}^{2} + \lambda\left(x - 2\right) = 0$.

$⇒ {x}^{2} + {y}^{2} + \left(\lambda - 4\right)x - 4y + \left(8 - 2\lambda\right) = 0$

Radius$= 3 = \sqrt{{\left(\dfrac{\lambda - 4}{2}\right)}^{2} + 4 - \left(8 - 2\lambda\right)}$

$⇒ 9 = \dfrac{{\lambda}^{2} - 8\lambda + 16}{4} - 4 + 2\lambda$

$⇒ 36 = {\lambda}^{2} - 8\lambda + 16 - 16 + 8\lambda$

$⇒ \lambda = \pm 6$

Hence, equation of the circle is ${x}^{2} + {y}^{2} - 10x - 4y + 20 = 0$.

So, $x$-intercept$= 2\sqrt{25 - 20} = 2\sqrt{5}$.