Example 18 (1993 National High School League Question) Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$. If $s=$ $x^{2}+y^{2}$, then $\frac{1}{s_{\max }}+\frac{1}{s_{\text {min }}}=$
Solve: Fill in $\frac{8}{5}$. Reason: From $x^{2}+y^{2}=s$, we think of $\sin ^{2} \theta+\cos ^{2} \theta=1$. Let's assume $x=\sqrt{s} \cos \theta, y=\sqrt{s} \sin \theta$. Substituting into the given equation $4 x^{2}-5 x y+4 y^{2}=5$, we get $4 s \cos ^{2} \theta-5 s \cos \theta \cdot \sin \theta+4 s \cdot \sin ^{2} \theta=5$, solving which yields $s=\frac{10}{8-5 \sin 2 \theta}$. Thus, $s_{\max }=\frac{10}{3}, s_{\min }=\frac{10}{13}$. Therefore, $\frac{1}{s_{\max }}+\frac{1}{s_{\min }}=\frac{8}{5}$.