I know that for any nonzero $x,y\in\mathbf{R}$,
$$ax^2+by^2+cxy > 0,$$
where $a,b,c\in\mathbf{R}$. What can I deduce about $a$, $b$, and $c$?
For example, letting $x=1$ and $y=0$, I know that $\boxed{a>0}$. Letting $x=0$ and $y=1$, I know that $\boxed{b>0}$. Letting $x=y=1$, I know that $\boxed{c>-(a+b)}$.
What else can I deduce about $a$, $b$, and $c$? How will I know when it's time to stop looking? (Is that last question even answerable?)
You can write $$ax^2+by^2+cxy=a\left(x+\frac{c}{2a}\,y\right)^2+\left(b-\frac{c^2}{4a}\right)y^2$$ by completing the square. This is a sum of two squares. In order to ensure that it's always positive, the coefficients both have to be positive, so $$a>0$$ and $$c^2<4ab.$$
Dividing the whole thing by $y^2$ , we get $a(x/y)^2+c(x/y)+b>0$ (Note: we can divide by $y^2$ since it is non zero) Since it is positive for all values of $(x/y)$, the discriminant must be negative (since $a$ is given to be positive) Therefore, $c^2 - 4 ab < 0$
