Circle$\left(s\right)$ touching $x-$axis at a distance $3$ from the origin and having an intercept of length $2\sqrt{7}$ on $y-$axis, is (are)

Let the equation of the circle be $x^2+y^2+2gx+2fy+c=0.$ It touches $x-$axis at a distance $3$ from the origin. Therefore, $c=g^2$ and the circle passes through $(\pm 3, 0).$

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}9\pm 6g+c=0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}9\pm 6g+g^2=0\Rightarrow (g\pm 3{)}^2=0\Rightarrow g=-3\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}c=g^2\Rightarrow c=9\end{array}$

The circle cuts an intercept of length $2\sqrt{7}$ on $y-$axis.

$\therefore 2\sqrt{f^2-c}=2\sqrt{7}\Rightarrow f^2-9=7\Rightarrow f=\pm 4$

Hence, the equations of the circles are

   $x^2+y^2-6x\pm 8y+9=0$