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This problem trains for: SAT-I, AMC-8, GMAT.

In a plane, two concentric circles (circles that have the same center) have radii with lengths 6 and 2. Two circles are tangent to both larger circles. What is the difference between the maximum and the minimum angles under which the centers of the smaller circles can be seen by an observer located at the center of the larger circles? (figure not to scale)

figure

The two smaller circles are the farthest apart when they are diametrically opposite:

figure

In this case, the angle under which their centers are seen is equal to 180°.

They are at the smallest possible distance when they are tangent to each other:

figure

Because the circles are tangent, their centers are collinear (on the same line) with the point of contact. This means that the distances are:

equation

equation

equation

and the triangle OPQ is equilateral. Therefore:

equation

and the difference of angles is:

equation