# 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 $c \in \R_{> 0}$.