I come across an optimization problem of the following form.

$$\max {\frac{1+v}{1-u}} \qquad \text{s.t.} \qquad ux^2+vy^2-xy \ge 0, \quad \forall x,y\in\mathbb{R}$$

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.

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