Value of Discriminant of Conic Section

Theorem

Let $K$ be a conic section embedded in a Cartesian plane with the general equation:

$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$

where $a, b, c, f, g, h \in \R$.

The value of the discriminant of $K$ is:

$\Delta = a b c + 2 f g h - a f^2 - b g^2 - c h^2$

By definition, the discriminant of $K$ is:

$\ds \Delta$

$=$

$\ds \begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}$

$\ds $

$=$

$\ds a \paren {b c - f^2} - h \paren {h c - g f} + g \paren {h f - g b}$

$\ds $

$=$

$\ds a b c - a f^2 - h^2 c + h g f + h g f - g^2 b$

$\ds $

$=$

$\ds a b c + 2 f g h - a f^2 - b g^2 - c h^2$

rearranging

$\blacksquare$