Let a circle C: (x - h) ^ 2 + (y - k) ^ 2 = r ^ 2 , k > 0 , touch...

Since the circle is touching $x$-axis so, $k = r$ and $h = 1$

Now by diagram, $OP = r, PR = 1$

$OR = \left|\dfrac{r + 1}{\sqrt{2}}\right|$

${r}^{2} = 1 + \dfrac{{\left(r + 1\right)}^{2}}{2}$

$2{r}^{2} = 2 + {r}^{2} + 1 + 2r$

${r}^{2} - 2r - 3 = 0$

\left(r - 3\right)\left(r + 1\right) = 0$</p><p>$r = 3, - 1$</p><p>$h + k + r = 1 + 3 + 3$</p><p>$= 7