Parametric Equations

Consider the ellipse with rectangular equation

    \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\]

If x=a\cos \theta and y=b\sin \theta, then

    \[\frac{x^2}{a^2}+\frac{y^2}{b^2}= \frac{(a\cos \theta)^2}{a^2}+\frac{(b\sin \theta)^2}{b^2}=\cos^2\theta+\sin^2\theta=1\]

and hence (x,y) lies on the ellipse.
The parametric equations of the ellipse are given by

    \[<span class="ql-right-eqno"> </span><span class="ql-left-eqno"> </span><img src="http://darel.hardycalculus.com/wp-content/ql-cache/quicklatex.com-7b1ba4301a9240993ef23dff1e603c30_l3.png" height="42" width="104" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{eqnarray*} x &=&a\cos \theta \\ y &=&b\sin \theta \end{eqnarray*}" title="Rendered by QuickLaTeX.com"/>\]

Ellipse4