Classification of Conic Sections by Coefficients of General Equation

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

Let $\Delta$ denote the discriminant of $K$:

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

If $\Delta \ne 0$, then $K$ is an ellipse, a parabola or a hyperbola, according to:

$\ds h^2 - a b$

$<$

$\ds 0:$

$K$ is an ellipse

$\ds h^2 - a b$

$=$

$\ds 0:$

$K$ is a parabola

$\ds h^2 - a b$

$>$

$\ds 0:$

$K$ is a hyperbola

If $\Delta = 0$, then $K$ is a degenerate conic:

$\ds h^2 - a b$

$<$

$\ds 0:$

$K$ is a point

$\ds h^2 - a b$

$=$

$\ds 0:$

$K$ is a pair of parallel lines or coincident straight lines, or an imaginary locus

$\ds h^2 - a b$

$>$

$\ds 0:$

$K$ is a pair of intersecting straight lines

Proof

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