Lower bound of a polynomial of degree 2

Question

I want to find out the conditions to guarantee the following:

Let $\alpha >0, \beta > 0,\gamma \geq 1$ and consider the polynomial of degree 2: $P(x,y) = \alpha^2 x^2 + \beta^2 y^2 - 2 \alpha \beta \gamma xy$ for $x\geq 0$ and $y\geq 0$.

Is it possible to lower bound $P(x,y)$ as follows:

$$ P(x,y) \geq C(Ax- By)^2 $$

for some positive constants $A, B, C$ depending on $\alpha, \beta,$ and $\gamma$?

Thank you in advance.

Answer 1

It is not possible for $\gamma>1$:

Note that $P(\beta,\alpha)=2\alpha^2\beta^2(1-\gamma)$.