# What is the parametric equation of an ellipse?

Apr 21, 2018

Here is one example...

#### Explanation:

You can have $\left(n \sin \left(t\right) , m \cos \left(t\right)\right)$ when $n \ne m$, and $n$ and $m$ do not equal to $1$.

This is essentially because:

$\implies x = n \sin \left(t\right)$

$\implies {x}^{2} = {n}^{2} {\sin}^{2} \left(t\right)$

$\implies {x}^{2} / {n}^{2} = {\sin}^{2} \left(t\right)$

$\implies y = m \cos \left(t\right)$

$\implies {y}^{2} / {m}^{2} = {\cos}^{2} \left(t\right)$

$\implies {x}^{2} / {n}^{2} + {y}^{2} / {m}^{2} = {\sin}^{2} \left(t\right) + {\cos}^{2} \left(t\right)$

Using the fact that ${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$...

$\implies {x}^{2} / {n}^{2} + {y}^{2} / {m}^{2} = 1$

This is essentially an ellipse!

Note that if you want a non-circle ellipse, you have to make sure that $n \ne m$