Example 1 Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=$ 5. If $s=x^{2}+y^{2}$, then $\frac{1}{s_{\text {max }}}+\frac{1}{s_{\text {min }}}=$ $\qquad$ .
(1993. National High School Mathematics Competition)
Analysis: From $x^{2}+y^{2}=s$, we associate it with $\sin ^{2} \theta+\cos ^{2} \theta$ $=1$. Let's set $x=\sqrt{s} \cos \theta, y=\sqrt{s} \sin \theta$, and substitute into the given equation to get
$4 s \cos ^{2} \theta-5 s \cos \theta \cdot \sin \theta+4 s \sin ^{2} \theta=5$.
Solving this, we get $s=\frac{10}{8-5 \sin 2 \theta}$.
Thus, $s_{\max }=\frac{10}{3}, s_{\min }=\frac{10}{13}, \frac{1}{s_{\max }}+\frac{1}{s_{\min }}=\frac{8}{5}$.