Question

The parametric representation of a point of the ellipse whose focii are $(3,0)$ and $(-1,0)$ and eccentricity $\frac{2}{3}$ is

Answer 1

We have

$2ae=4$ .............................(Distance between 2 focii)

$e=\frac{2}{3}$

So

$a=3$ Then

$a^2{e}^2=a^2-b^2$

Hence $b=\sqrt{9(1-\frac{4}{9})}=\sqrt{5}$

So parametric representation of a point on an ellipse is

$(3\cos t,\sqrt{5}\sin t)$

Where t is angle made by projection of point on the auxillary circle.