Circle(s) touching x -axis at a distance 3 from the origin and having an intercept of length 2 7 on y -axis…

Let  the circle be

$x^{2 }+ y^2 + 2gx + 2fy + c = 0$

According to the question, 

$g^2-c=0\Rightarrow g^2=c ....\left(i\right)$ 

and     

$2\sqrt{f^2-c}=2\sqrt{7}\Rightarrow f^2-c=7.....\left(ii\right)$

The circle passes through the point $\left(3, 0\right)$, then 
$9+0+6g+0+c=0$
$\Rightarrow 9+6g+g^2=0,$ by $\left(i\right)$
$\Rightarrow (g+3{)}^2=0\Rightarrow g=-3$
$\therefore$$c = 9$ and $f^2 = 16 \Rightarrow f=\pm 4$     

So the required circles are

 $x^2+y^2-6x\pm 8y+9=0$.