16. 7 (US MO 16) In the plane, there are three circles $C_{i}(i=1,2,3)$, where the diameter of $C_{1}$ is $A B=1$; $C_{2}$ is concentric with $C_{1}$, has a diameter of $k$, and satisfies $1<k<3$; $C_{3}$ has $A$ as its center and $2 k$ as its diameter ($k$ is a constant). Consider all line segments $X Y$, one end $X$ on $C_{2}$, the other end $Y$ on $C_{3}$, and $X Y$ contains point $B$. For what ratio $X B: B Y$ is the length of segment $X Y$ minimized?
16.7 Let $\angle A B X=\alpha, \angle A Y B=\beta, \angle B X O=\gamma, O B=\frac{1}{2}, O X=\frac{1}{2} k, A Y=k$.

In $\triangle A B Y$ and $\triangle O B X$, by the Law of Sines, we can deduce that $\beta=\gamma$ and $X Y=X B+B Y=\frac{1}{2}(3 k \cos \gamma-\cos \alpha)$, where $k=\frac{\sin \alpha}{\sin \gamma}$.
Draw $A D \perp B Y$ at $D$, then $Y D=k \cos \gamma, A D=\sin \alpha$.
Let $\sin ^{2} \alpha=x$, we can transform it into finding the minimum value of $2 X Y=u=3 \sqrt{k^{2}-x}-\sqrt{1-x}$. Squaring both sides and transforming it into a quadratic equation, let $\triangle \geqslant 0$, we get $u \geqslant 2 \sqrt{2} \sqrt{k^{2}-1}$. When $x=\frac{9-k^{2}}{8}$, the equality holds, $X Y=\sqrt{2} \sqrt{k^{2}-1}$ is the minimum value. When calculating the value of $X B: B Y$ at this time, the value is 1, which makes $X Y$ the shortest.