I come across a problem like

$\max {\frac{1+v}{1-u}}$

$s.t.~$ $ux^2+vy^2-xy\ge0$ $\forall x,y\in\mathbb{R}$

I do not know much of optimization.
What I have done is that $ux^2+vy^2\ge 2\sqrt{uv}xy\ge xy$, so I let $uv=\frac{1}{4}$ and get the seemingly correct answer.

My question is that What kind of opt problem is it? Where can I get some resources to quickly learning to solving this kind of problem?