Circle(s) touching x -axis at a distance 3 from the origin and...

Let the circle be

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

According to the question,

${g}^{2} - c = 0 ⇒ {g}^{2} = c . . . . \left(i\right) </p><p>and </p><p>$2\sqrt{{f}^{2} - c} = 2\sqrt{7} ⇒ {f}^{2} - c = 7 . . . . . \left(ii\right)$</p><p>The circle passes through the point \left(3, 0\right)$, then
$9 + 0 + 6g + 0 + c = 0$
$⇒ 9 + 6g + {g}^{2} = 0,$ by \left(i\right)$<br />$⇒ ( g + 3{)}^{2} = 0 ⇒ g = - 3$<br />∴c = 9 and${f}^{2} = 16 ⇒ f = \pm 4$</p><p>So the required circles are</p><p>${x}^{2} + {y}^{2} - 6x \pm 8y + 9 = 0$.