2. Let positive numbers $a, b$ satisfy $a<b$. If there are real numbers $x_{1}, y_{1}, x_{2}, y_{2}$ such that $x_{1}+y_{1}, x_{2}+y_{2}$ are the arithmetic mean and geometric mean of $a$ and $b$ respectively. Then the maximum value of $\frac{\sqrt{x_{1} y_{1}}}{\left(x_{2}+y_{2}\right)^{2}}$ is ( ).
(A) $\frac{\sqrt{a b}}{(a+b)^{2}}$
(B) $\frac{\sqrt{a+b}}{(a b)^{2}}$
(C) $\frac{4 \sqrt{a b}}{(a+b)^{2}}$
(D) $\frac{a+b}{16 \sqrt{a b}}$
2. D.

Given $x_{1}+y_{1}=\frac{a+b}{2}, x_{2} y_{2}=\sqrt{a b}(0<a<b)$, then
$$
\frac{\sqrt{x_{1} y_{1}}}{\left(x_{2}+y_{2}\right)^{2}} \leqslant \frac{\frac{x_{1}+y_{1}}{2}}{4 x_{2} y_{2}}=\frac{a+b}{16 \sqrt{a b}} .
$$