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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