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 $\qquad$
Let $S=x^{2}+y^{2} \Rightarrow x=\sqrt{S} \cos \theta, y=\sqrt{S} \sin \theta$,
then $x^{2}-2 x y+5 y^{2}=S\left(\cos ^{2} \theta-2 \sin \theta \cos \theta+5 \sin ^{2} \theta\right)=S(3-2 \cos 2 \theta-\sin 2 \theta)=4$ $\Rightarrow 3-\sqrt{5} \sin (2 \theta+\varphi)=\frac{4}{S} \in[3-\sqrt{5}, 3+\sqrt{5}] \Rightarrow S \in[3-\sqrt{5}, 3+\sqrt{5}]$