1. Let real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$. If $S=x^{2}+y^{2}$, denote the maximum and minimum values of $S$ as $p$ and $q$, respectively, then $\frac{1}{p}+\frac{1}{q}=$ $\qquad$
$-1 . \frac{8}{5}$.
Let's assume $x=\sqrt{S} \cos \theta, y=\sqrt{S} \sin \theta$.
Substituting into the equation in the problem, we get
$$
\begin{array}{l}
S=\frac{5}{4-\frac{5}{2} \sin 2 \theta} \Rightarrow p=\frac{10}{3}, q=\frac{10}{13} \\
\Rightarrow \frac{1}{p}+\frac{1}{q}=\frac{8}{5} .
\end{array}
$$