Example 3 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 $\frac{1}{s_{\text {max }}}+\frac{1}{s_{\text {min }}}$ is $\qquad$ (1993, National High School Mathematics Competition)
Solution: From the given, we know that $(x y)^{2}=\left(\frac{4}{5} s-1\right)^{2}$.
Also, $x^{2}+y^{2}=s$,
$\therefore x^{2}, y^{2}$ are the two real roots of the equation $t^{2}-s t+\left(\frac{4}{5} s-1\right)^{2}=0$.
$$
\begin{aligned}
\therefore \Delta & =s^{2}-4\left(\frac{4}{5} s-1\right)^{2} \\
& =-\frac{39}{25} s^{2}+\frac{32}{5} s-4 \geqslant 0 .
\end{aligned}
$$

Solving this, we get $\frac{10}{13} \leqslant s \leqslant \frac{10}{3}$.
Therefore, $\frac{1}{s_{\max }}+\frac{1}{s_{\min }}=\frac{8}{5}$.