Equation of Conic Section

Theorem

Cartesian Form

The general conic section can be expressed in Cartesian coordinates in the form:

$a x^2 + b x y + c y^2 + d x + e y + f = 0$

for some $a, b, c, d, e, f \in \R$.

Cartesian Form using Eccentricity

Let $K$ be a conic section embedded in a Cartesian plane such that:

one focus of $K$ is at the origin

the eccentricity of $K$ is $e$

the directrix of $K$ is a distance $h$ from the origin.

Then $K$ can be described using the equation:

$\paren {1 - e^2} x^2 + 2 e^2 h x + y^2 = e^2 h^2$

Discriminant Form

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

Then after translation of coordinate axes, $K$ can be described using the equation:

$a x^2 + 2 h x y + b y^2 - \dfrac \Delta {h^2 - a b} = 0$

where $\Delta$ is the discriminant of $K$:

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