8. Real numbers $x, y$ satisfy $4 x^{2}+5 x y+4 y^{2}=5$, let $S=x^{2}+y^{2}, S_{\max }$ ・
$$
S_{\min }=
$$
$\qquad$
8. $\frac{100}{39}$.

First, $x=\sqrt{S} \cos \theta, y=\sqrt{S} \sin \theta, 4 S+5 S \cdot \sin \theta \cos \theta=5, S=\frac{5}{4+\frac{5}{2} \sin 2 \theta}$. When $\sin 2 \theta=-1$, $S_{\text {max }}=\frac{10}{3}$; when $\sin 2 \theta=1$, $S_{\text {min }}=\frac{10}{13}, S_{\text {max }} \cdot S_{\text {min }}=\frac{100}{39}$.