# Thread: find the parametric equation of an ellipse?

1. ## find the parametric equation of an ellipse?

how to find the parametric equation of an ellipse?

x^2/a^2 + y^2/b^2 =1

thank you very much.

2. Originally Posted by kittycat
how to find the parametric equation of an ellipse?

x^2/a^2 + y^2/b^2 =1

thank you very much.
the ellipse $\displaystyle \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$ is given by the parametric equations: $\displaystyle x = a \cos t \mbox{ , } y = b \sin t$ for $\displaystyle 0 \le t \le 2 \pi$

3. hi Jhevon,

How do you work out this parametric equation?

please teach me . thank you.

4. Originally Posted by kittycat
hi Jhevon,

How do you work out this parametric equation?

please teach me . thank you.
we know that $\displaystyle \cos^2 t + \sin^2 t = 1$

and we also know from our knowledge of polar coordinates that it is more appropriate to relate the x to the cosine and the y to the sine, so

$\displaystyle \frac {x^2}{a^2} + \frac {y^2}{b^2} = \left( \frac xa \right)^2 + \left( \frac yb \right)^2 = 1$

thus we can let $\displaystyle \cos t = \frac xa$ and $\displaystyle \sin t = \frac yb$ (and so we would obtain $\displaystyle \cos^2 t + \sin^2 t = 1$)

solving for $\displaystyle x$ and $\displaystyle y$ we get: $\displaystyle x = a \cos t$ and $\displaystyle y = b \sin t$ and we restrict $\displaystyle t$ to $\displaystyle 0 \le t \le 2 \pi$ since sine and cosine are periodic for that interval

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# derivation of the parametric equation of ellipse

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