Question

The coordinates of a point, in the parametric form,on the Ellipse whose foci are $(-1,0)$ and $(7,0)$ and $e=1/2$ are

• A. $(8\cos \theta ,4\sqrt{3}\sin \theta )$
• B. $(3+8\cos \theta ,4\sqrt{3}\sin \theta )$
• C. $(3+4\sqrt{3}\cos \theta ,8\sin \theta )$
• D. $(3+4\cos \theta ,2\sqrt{3}\sin \theta )$

Answer 1

The correct option is B $(3+8\cos \theta ,4\sqrt{3}\sin \theta )$

Given foci are $(-1,0)$ and $(7,0)$ . $e=\frac{1}{2}$

center of ellipse=$=(\frac{-1+7}{2},\frac{0+0}{2})=(3,0)$

foci are on the axis. so, distance between center and focus $=ae$

$\Rightarrow ae=4$

$\Rightarrow a=8$

$e=\frac{1}{2}\Rightarrow \sqrt{\frac{a^2-b^2}{a^2}}=\frac{1}{2}$

$\Rightarrow \sqrt{64-b^2}=4$

$\Rightarrow b^2=48\Rightarrow b=4\sqrt{3}$

If $(h,k)$ is the center of an ellipse with semimajor axis as $a$ and $b$, Parametric form would be $(h+a\cos \theta ,k+b\sin \theta )$. $\therefore$ parametric form$=(3+8\cos \theta ,4\sqrt{3}\sin \theta )$