# A different kind of pursuit .....

Geometry Level 3

Consider two concentric circles, radii $$1$$ and $$2$$ respectively, centered at the origin. Particles are situated on each of the circles, at $$(1,0)$$ and $$(2,0)$$ respectively, and then simultaneously begin to move counterclockwise around their respective circles, both at a rate of $$1$$ unit per second.

The time that has elapsed when the line joining the two particles is tangent to the smaller circle for the first time is $$\dfrac{a\pi}{b}$$ seconds, where $$a$$ and $$b$$ are positive coprime integers. Find $$a + b$$.

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