3. Given real numbers $x_{1}, x_{2}, y_{1}, y_{2}$ satisfy
$$
\begin{array}{l}
x_{1}^{2}+25 x_{2}^{2}=10, \\
x_{2} y_{1}-x_{1} y_{2}=25, \\
x_{1} y_{1}+25 x_{2} y_{2}=9 \sqrt{55} .
\end{array}
$$

Then $y_{1}^{2}+25 y_{2}^{2}=$ $\qquad$
3.2008 .

Notice that
$$
\begin{array}{l}
\left(x_{1}^{2}+a x_{2}^{2}\right)\left(y_{1}^{2}+a y_{2}^{2}\right) \\
=\left(x_{1} y_{1}+a x_{2} y_{2}\right)^{2}+a\left(x_{2} y_{1}-x_{1} y_{2}\right)^{2},
\end{array}
$$

where $a$ is any real number.
For this problem, take $a=25$, and it is easy to find that $y_{1}^{2}+25 y_{2}^{2}=2008$.