Equation of Ellipse in Reduced Form

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 $K$.

Also see

• Equation of Hyperbola in Reduced Form