3. The cross-sectional area of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ through the diagonal $B D_{1}$ is $S$, and $S_{1}$ and $S_{2}$ are the maximum and minimum values of $S$, respectively. Then $\frac{S_{1}}{S_{2}}$ is ( ).
(A) $\frac{\sqrt{3}}{2}$
(B) $\frac{\sqrt{6}}{2}$
(C) $\frac{2 \sqrt{3}}{3}$
(D) $\frac{2 \sqrt{6}}{3}$
(Answer: (C).)