"What is the parametric equation of an ellipse?" solved and explained

Isabelle Diaz

Answered question

2022-03-15

What is the parametric equation of an ellipse?

Brock Floyd

Beginner2022-03-16Added 6 answers

Step 1
You can have

(n \sin (t), m \cos (t))

when

n \neq m

, and n and m do not equal to 1.
This is essentially because:

\Rightarrow x = n \sin (t)

\Rightarrow x^{2} = n^{2} \sin^{2} (t)

\Rightarrow \frac{x^{2}}{n^{2}} = \sin^{2} (t)

\Rightarrow y = m \cos (t)

\Rightarrow \frac{y^{2}}{m^{2}} = \cos^{2} (t)

\Rightarrow \frac{x^{2}}{n^{2}} + \frac{y^{2}}{m^{2}} = \sin^{2} (t) + \cos^{2} (t)

Step 2
Using the fact that

\sin^{2} (x) + \cos^{2} (x) = 1

\Rightarrow \frac{x^{2}}{n^{2}} + \frac{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 \neq m

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