2 If the ellipse $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}=1(m, n>0)$ passes through the point $p(a, b)(a b \neq 0,|a| \neq|b|)$, find the minimum value of $m+n$.

The ellipse $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}=1(m, n>0)$ passes through the point $p(a, b)(a b \neq 0,|a| \neq|b|)$. Determine the minimum value of $m+n$.
2. Let $m, n \in \mathbf{R}^{+}, a, b \in \mathbf{R}^{+}$, and set $a=m \cos \alpha, b=n \sin \alpha$, where $\alpha \in \left(0, \frac{\pi}{2}\right]$. Then,
$$(m+n)^{2}=\left(\frac{a}{\cos \alpha}+\frac{b}{\sin \alpha}\right)^{2}=\frac{a^{2}}{\cos ^{2} \alpha}+\frac{b^{2}}{\sin ^{2} \alpha}+\frac{2 a b}{\sin \alpha \cos \alpha}=a^{2}\left(1+\tan ^{2} \alpha\right)+b^{2}\left(1+\frac{1}{\tan ^{2} \alpha}\right)+\frac{2 a b\left(1+\tan ^{2} \alpha\right)}{\tan \alpha}.$$

Let $\tan \alpha=t$, then $t \in \mathbf{R}^{+}$, hence
$$\begin{aligned}
(m+n)^{2} & =\left(a^{2}+b^{2}\right)+a^{2} t^{2}+\frac{b^{2}}{t^{2}}+\frac{2 a b}{t}+2 a b t \\
& =a\left(a t^{2}+\frac{b}{t}+\frac{b}{t}\right)+b\left(\frac{b}{t^{2}}+a t+a t\right)+\left(a^{2}+b^{2}\right) \\
& \geqslant a \cdot 3 \sqrt[3]{a b^{2}}+b \cdot 3 \sqrt[3]{a^{2} b}+\left(a^{2}+b^{2}\right)
\end{aligned}$$

Therefore, $(m+n)_{\min }=\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}\right)^{\frac{3}{2}}$, with equality when $t=\sqrt[3]{\frac{b}{a}}$.
Note: The problem can also be solved using the Cauchy inequality.
Since $\left(a^{\frac{1}{3}} \cos \alpha+b^{\frac{1}{3}} \sin \alpha\right)^{2} \leqslant\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}\right)\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)$, we have $\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}\right)^{\frac{1}{2}}\left(\frac{a}{\cos \alpha}+\frac{b}{\sin \alpha}\right) \geqslant\left(a^{\frac{1}{3}} \cos \alpha+b^{\frac{1}{3}} \sin \alpha\right)\left(\frac{a}{\cos \alpha}+\frac{b}{\sin \alpha}\right) \geqslant\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}\right)^{2}$, i.e., $m+n \geqslant\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}\right)^{\frac{2}{2}}$, and equality holds when $\alpha=\arctan \sqrt[3]{\frac{a}{b}}$.