Question

The parametric representation of a point on the ellipse whose foci are $(-1,0)$ and $(7,0)$ and eccentricity $1/2$ is

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

Answer 1

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

Distance between two foci, $2ae=7+1=8$

$\therefore$ $ae=4$

$\Rightarrow$ $a=8$ ($\because e=\frac{1}{2}$ given)

Now, $b^2=a^2(1-e^2)=64(1-\frac{1}{4})$

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

Since, the centre of the ellipse is the midpoint of the line joining two foci,therefore the coordinates of the centre are $(3,0)$.

In equation is

$\frac{{(x-3)}^2}{8^2}+\frac{{(y-0)}^2}{{\left(4\sqrt{3}\right)}^2}=\mathrm{1......}\left(i\right)$

hence the parametric coordinates of a point on Eq.$\left(i\right)$ are $(3+8\cos \theta ,4\sqrt{3}\sin \theta )$