1. If real numbers $x, y$ satisfy $x^{2}-2 x y+5 y^{2}=4$, then the range of $x^{2}+y^{2}$ is $\qquad$ .
$-1 \cdot[3-\sqrt{5}, 3+\sqrt{5}]$.
According to the problem, we have $(x-y)^{2}+(2 y)^{2}=4$.
Let $x-y=2 \cos \theta, y=\sin \theta$.
Thus, $x=\sin \theta+2 \cos \theta, y=\sin \theta$.
Then, $x^{2}+y^{2}=(\sin \theta+2 \cos \theta)^{2}+\sin ^{2} \theta$ $=3+\sqrt{5} \sin (2 \theta+\varphi)$.
Therefore, the range of $x^{2}+y^{2}$ is $[3-\sqrt{5}, 3+\sqrt{5}]$.