Three half circles and one mini circle

Solution by: Nibedan Norman Mukherjee

Let the radius of circle with center $P$ be $r$ and the center of the biggest semicircle be $O(0,0)$ . Then $A(-2,0)$ and $B(3,0)$ and point $P$ satisfies the following three equations of circle:

$\begin{cases} (x+2)^2 + y^2 = (3+r)^2 & ...(1) \\ (x-3)^2 + y^2 = (2+r)^2 & ...(2) \\ x^2 + y^2 = (5-r)^2 & ...(3) \end{cases}$

$\begin{cases} x^2 + 4x+4 + y^2 = 9 + 6r+ r^2 & ...(1) \\ x^2 - 6x+9 + y^2 = 4 + 4r+ r^2 & ...(2) \\ x^2 + y^2 = 25 -10 r+ r^2 & ...(3) \end{cases}$

$\begin{cases} (1) - (3): & 4 x + 4 = - 16 + 16r & \implies x = 4r - 5 & ...(4) \\ (3)-(2): & 6x - 9 = 21 - 14 r & \implies 3x = 15 - 7r & ...(5) \end{cases}$

$\begin{aligned} 3\times (4) - (5): \quad 19r - 30 & = 0 \\ \implies r & = \frac {30}{19} \end{aligned}$

Therefore, $a+b = 19+30 = \boxed{49}$ .