6. As shown in the figure, in two circles with $O$ as the center, $A$ is any point on the larger circle, and a secant line $A X Y$ is drawn through $A$ to the smaller circle. If $A X \cdot A Y = 3$, then the area of the annulus in the figure is ( .
(A) $3 \pi$
(B) $6 \pi$
(C) $9 \pi$
(D) cannot be determined
6. (A).

As shown in the figure, draw the tangent $AB$ of the small circle through $A$, with the point of tangency being $B$, then
$$
AB^2 = AX \cdot AY = 3
$$

Connect $2U, \cup B$, then
$$
\begin{array}{c}
S_{\text {quadrilateral}} = \pi \cdot OA^2 - \pi \cdot OB^2 \\
= \pi \left(OA^2 - OB^2\right) = \pi \cdot AB^2 = 3\pi .
\end{array}
$$