I was thinking about the above problem .Can someone point me in the right direction? Thanks in advance for your time.
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For any point $A$ on the circle, $B$ will be any point on the circle of radius $b$ with center $A$. Since this will be true for any $A$ on the circle, the locus of $B$ will be the figure formed by a circle that rotates around a point. Therefore, I would say the right answer is C, since for any point in the region between the two cocentric circles you can find an $A$ such that $A$ is on the circle and $|AB|=b$.
The locus of B takes every point on the circle to another, some distance $b$ from the original in a given direction. Does that help?