8. Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$, let $S=x^{2}+y^{2}$, then the value of $\frac{1}{S_{\max }}+\frac{1}{S_{\text {min }}}$ is
$$
\begin{array}{l}
\text { Let } x=\sqrt{S} \cos \theta, y=\sqrt{S} \sin \theta \Rightarrow S\left(4 \cos ^{2} \theta-5 \cos \theta \sin \theta+4 \sin ^{2} \theta\right)=5 \\
\Rightarrow 4-\frac{5}{2} \sin 2 \theta=\frac{5}{S} \in\left[\frac{3}{2}, \frac{13}{2}\right] \Rightarrow \frac{1}{S} \in\left[\frac{3}{10}, \frac{13}{10}\right] \Rightarrow \frac{1}{S_{\max }}+\frac{1}{S_{\min }}=\frac{8}{5} .
\end{array}
$$