If a circle of radius 2 touches X-axis at $\left(1,0\right)$  then its centre may be

Let $x^2+y^2+2gx+2fy+c=0$  be the circle

Since the circle touches x-axis at $\left(1,0\right)$  then

$g^2=c$  and $r=\left|f\right|$ .

$2=\left|f\right|$

$\Rightarrow f=\pm 2$

Since circle passes through $\left(1,0\right)$  then

$\Rightarrow 1+2g+c=0$

$\Rightarrow 1+2g+g^2=0$  $\left(\because c=g^2\right)$

$\Rightarrow {\left(g+1\right)}^2=0$

$\Rightarrow g=-1$

$\therefore$ centre$=\left(-g,-f\right)=\left(1,\pm 2\right)$

$\therefore$ centre is $\left(1,2\right)$  or $\left(1,-2\right)$