Definition:Discriminant of Quadratic Equation in Two Variables

Definition
Consider the quadratic equation in $2$ variables:


 * $(1): \quad a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$

where $x$ and $y$ are independent variables.

The discriminant of $(1)$ is the expression:


 * $a b c + 2 f g h - a f^2 - b g^2 - c h^2$

Also presented as
It can also be expressed in the form of a determinant:
 * $\begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}$