Definition:Conic Section/Reduced Form
Definition
Let $K$ be a conic section.
Let $K$ be embedded in a cartesian plane such that:
for some $c \in \R_{\ge 0}$.
This can be interpreted in the contexts of the specific classes of conic section as follows:
Reduced Form of Ellipse
Let $K$ be an ellipse embedded in a cartesian plane.
$K$ is in reduced form if and only if:
- $(1)$ its major axis is aligned with the $x$-axis
- $(2)$ its minor axis is aligned with the $y$-axis.
Reduced Form of Hyperbola
Let $K$ be a hyperbola embedded in a cartesian plane.
$K$ is in reduced form if and only if:
- $(1)$ its major axis is aligned with the $x$-axis
- $(2)$ its minor axis is aligned with the $y$-axis.
Reduced Form of Circle
Let $K$ be a circle embedded in a cartesian plane.
$K$ is in reduced form if and only if its center is located at the origin.
Reduced Form of Parabola
Let $K$ be a parabola embedded in a cartesian plane.
As a Parabola has no Center, it is not possible to define the reduced form of a parabola in the same way as for the other classes of conic section.
Instead, $K$ is in reduced form if and only if:
- $(1)$ its focus is at the point $\tuple {c, 0}$
- $(2)$ its directrix is aligned with the line $x = -c$
for some $c \in \R_{> 0}$.