Category:Equation of Conic Section
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This category contains pages concerning Equation of Conic Section:
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 $K$:
- $\delta = \begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}$
Pages in category "Equation of Conic Section"
The following 5 pages are in this category, out of 5 total.