It's all Circles and Trapezoids.

For smaller circle:

$(a + b - 2r)^2 = (2r)^2 + (b - a)^2 \implies a^2 + 2ab + b^2 - 4(a + b)r + 4r^2 = 4r2 + b^2 - 2ab + a^2 \implies 4ab = 4(a + b)r^2$

$\implies \boxed{r = \dfrac{ab}{a + b}}$

For outer circle $(x - x_{0})^2 + (y - y_{0})^2 = R^2$ .

Using the three points on the larger circle I chose above:

$(0,0): x_{0}^2 + y_{0}^2 = R^2$

$(0,2r): x_{0}^2 + 4r^2 - 4ry_{0} + y_{0}^2 = R^2 \implies 4r(r - y_{0}) = 0 \:\ r \neq 0 \implies y_{0} = r = \dfrac{ab}{a + b}$

$(b,0):b^2 - 2bx_{0} + x_{0}^2 + y_{0}^2 = R^2 \implies x_{0} = \dfrac{b}{2}$

$\implies \boxed{R^2 = \dfrac{b^2}{4} + \dfrac{a^2b^2}{(a + b)^2}}$

The height of the trapezoid $h = 2r = \dfrac{2ab}{a + b} \implies$

$R^2 - h = \dfrac{b^{2}}{4} + \dfrac{a^2b^2}{(a + b)^2} - \dfrac{2ab}{a + b} =$ $\dfrac{5a^2b^2 + 2ab^3 + b^4 - 8a^2b - 8ab^2}{4(a + b)^2} = \dfrac{2ab^3 + b^4}{4(a + b)^2} \implies$ $ab(5ab - 8a - 8b) = 0 \:\ a,b \neq 0 \implies 5ab = 8(a + b) \implies r = \dfrac{ab}{a + b} = \dfrac{8}{5} \implies A = \dfrac{64}{25}\pi = \dfrac{\alpha}{\beta}\pi \implies \alpha + \beta = \boxed{89}$ .