<s>[INST] Basic inequality problem For $a,b,m,n\in \mathbb{R}$, such that $m^2n^2>a^2m^2+b^2n^2$, what is the relationship between $M=\sqrt{m^2+n^2}$ and $N=a+b$?
I deduced the following: $$m^2n^2>a^2m^2+b^2n^2\geq 2abmn$$, $m,n$ is nonzero from the given inequality, so $mn\geq 2ab$. However, I fail to draw any conclusion between $M$ and $N$, I knew the following relation $\sqrt{m^2+n^2}\leq |m|+|n|$ and the AM-GM inequality. Anyone have ideas?
 [/INST] Geometrically, $(a,b)$ lies on the $(x,y)$-plane inside the ellipse
$$
\frac{x^2}{n^2} + \frac{y^2}{m^2}=1.\tag1
$$
Since $a+b$ is constant on lines with slope $-1$, the value of $a+b$ is no greater than $A+B$, where $(A,B)$ is the point in the first quadrant on the ellipse where the slope of the tangent line is $-1$. Using implicit differentiation (are you allowed to use calculus?) the slope of the tangent to (1) at $(x,y)$ is $$-\frac{m^2}{n^2}\frac xy\tag2$$
so $A$ and $B$ satisfy the two equations
$$
\frac{A^2}{n^2} + \frac{B^2}{m^2}=1\qquad\text{and}\qquad
-\frac{m^2}{n^2}\frac AB = -1.
$$
Solving these simultaneously yields, after some algebra,
$$
a+b<A+B = \sqrt{m^2+n^2}.
$$
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