Shocked Face

Geometry Level 5

Let RR be the region in the first quadrant outside of the circles x2+y2=4x^{2} + y^{2} = 4 and x2+(y3)2=1x^{2} + (y - 3)^{2} = 1 but inside the circle x2+(y1)2=9.x^{2} + (y - 1)^{2} = 9.

The largest circle that can be inscribed in RR has radius ab\dfrac{a}{b}, where aa and bb are positive coprime integers. Find a+ba + b.

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