Assume that $f_0,f_1,f_2$ are polynomial functions of degree two in two variables. This means that the $f_i$ are linear combinations with real coefficients of $x^2,xy,x,y^2,y,1$.

Consider the function $f = f_1^2-af_0f_2:\mathbb{R}^2\rightarrow\mathbb{R}$ where $a\in \mathbb{R}_{>0}$. Is it true that for a "random" choice (whit a suitable definition of random) of the $f_i$ there exists $(x_0,y_0)\in\mathbb{R}^2$ such that $f(x_0,y_0) \geq 0$.

Clearly, this does not hold for any choice of the $f_i$. Take for instance $f_1\equiv 0$, $f_0 = x^2+1$, $f_2 = y^2+1$. Then $f(x,y) = -a(x^2y^2+x^2+y^2+1) < 0 $ for all $(x,y)\in\mathbb{R}^2$.

Thank you very much.