# Equation of Ellipse in Reduced Form

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## Theorem

Let $K$ be an ellipse aligned in a cartesian plane in reduced form.

Let:

- the major axis of $K$ have length $2 a$
- the minor axis of $K$ have length $2 b$.

### Cartesian Frame

The equation of $K$ is:

- $\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

### Parametric Form

The equation of $K$ in parametric form is:

\(\ds x\) | \(=\) | \(\ds a \cos \theta\) | ||||||||||||

\(\ds y\) | \(=\) | \(\ds b \sin \theta\) |

where $\theta$ is the eccentric angle of the point $P = \tuple {x, y}$ with respect to $K$.