A circle touching the x - axis at ( 3, 0 ) and making an intercept...

Since circle touches the x-axis at $\left(3, 0\right),$ hence centre can be taken as $\left(3, \alpha\right)$ and radius $= \left|\alpha\right|<br />Since y-intercept is 8, by applying Pythagorean theorem<br />${3}^{2} + {4}^{2} = {\alpha}^{2} ⇒ \alpha = 5, - 5$<br />Hence two circles are possible<br />${\left(x - 3\right)}^{2} + {\left(y - 5\right)}^{2} = 25 and {\left(x - 3\right)}^{2} + {\left(y + 5\right)}^{2} = 25 .
Clearly$( 3, 10 )$ satisfies the given equation of circle.