Question

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

• A. $ (1+3\text{ }cos\text{ }\theta ,\text{ }\sqrt{3}sin\theta ) $
• B. $ (1+3\text{ }cos\theta ,\text{ }5\text{ }sin\theta ) $
• C. $ (1+3\text{ }cos\theta ,1+\sqrt{5}sin\theta ) $
• D. $ (1+3\text{ }cos\theta ,1+5\text{ }sin\theta ) $
• E. $ (1+3cos\theta ,\sqrt{5}sin\theta ) $

Answer 1

Answer: (E)

Given that, foci are (3, 0) and $ (-1,0) $ and $ e=\frac{2}{3} $
$ \therefore $ $ 2ae=4 $
$ \Rightarrow $ $ 2\times a\times \frac{2}{3}=4\Rightarrow a=3 $ Also, $ e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}\Rightarrow \frac{4}{9}=1-\frac{{{b}^{2}}}{9} $
$ \Rightarrow $ $ \frac{{{b}^{2}}}{9}=1-\frac{4}{9}=\frac{5}{9}\Rightarrow b=\sqrt{5} $
$ \therefore $ Parametric representation of a point is
$ (1+3\cos \theta ,\sqrt{5}\sin \theta ). $