Question

The parametric representation of a point on the ellipse whose foci are $\left(-1,0\right)$ and $\left(7,0\right)$ and eccentricity $\frac{1}{2}$ is

• A. $\left(3+8\cos \theta ,4\sqrt{3}\sin \theta \right)$
• B. $\left(8\cos \theta ,4\sqrt{3}\sin \theta \right)$
• C. $\left(3+4\sqrt{3}\sin \theta ,8\sin \theta \right)$
• D. None of the above

Answer 1

The correct option is A

$\left(3+8\cos \theta ,4\sqrt{3}\sin \theta \right)$

Explanation for the correct option:

Step 1: Calculate the distance between foci.

It is given that the coordinates of the foci are $\left(-1,0\right)$ and $\left(7,0\right)$.

And, the eccentricity of the ellipse $\left(e\right)=\frac{1}{2}$

Now, we know that the distance $\left(d\right)$ between two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is calculated by,

$d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

So the distance between the foci $\left(-1,0\right)$ and $\left(7,0\right)$ is,

The distance between the foci $=\sqrt{{\left(7+1\right)}^2+{\left(0+0\right)}^2}$

$\Rightarrow$ The distance between the foci $=\sqrt{64}$

$\Rightarrow$ The distance between the foci $=8$ $...\left(\mathrm{i}\right)$

Step 2: Calculate the value of the major axis $a$.

Now, we know that,

The distance between the foci $=2ae$ [Here, $a$ is the major axis]

$\Rightarrow$ $8=2ae$

$\Rightarrow$ $8=2a\times \frac{1}{2}$ $\left[\because e=\frac{1}{2}\right]$

$\Rightarrow$ $8=a$

Step 3: Calculate the value of the minor axis $b$.

As we know that in an ellipse, the eccentricity $e$ is given by,

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

$\begin{array}{cl}\Rightarrow & b=a\sqrt{1-e^2}\\ \Rightarrow & b=8\times \sqrt{1-{\left(\frac{1}{2}\right)}^2}\\ \Rightarrow & b=8\times \sqrt{\frac{3}{4}}\\ \Rightarrow & b=8\times \frac{\sqrt{3}}{2}\\ \Rightarrow & b=4\sqrt{3}\end{array}$

Step 4: Calculate the coordinates of the center of the ellipse.

We know that, by the mid-point formula,

The coordinates of the mid-point of a line are obtained by joining two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is calculated by,

Mid-point $=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Since the center of an ellipse is the mid-point of the line joining the two foci of the ellipse.

And, the foci of the given ellipse are $\left(-1,0\right)$ and $\left(7,0\right)$.

So, the coordinates of the center $=\left(\frac{-1+7}{2},\frac{0+0}{2}\right)$

$\Rightarrow$ the coordinates of the center $=\left(\frac{6}{2},\frac{0}{2}\right)$

$\Rightarrow$ the coordinates of the center $=\left(3,0\right)$

Step 5: Form the equation of the given ellipse.

As we know that the equation of an ellipse with center at $\left(h,k\right)$ and major and minor axis at $a$ and $b$ respectively is given by,

$\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$

Since, for the given ellipse,

$\left(h,k\right)=\left(3,0\right)$

$a=8$

$b=4\sqrt{3}$

So, the equation of the given ellipse is,

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

Step 6: Deduce the parametric representation.

Now as we know the parametric equations of the ellipse are,

$x=a\cos \theta$ and $y=b\sin \theta$

So, $x=a\cos \theta$

$\Rightarrow$ $x-3=8\cos \theta$

$\Rightarrow$ $x=3+8\cos \theta$

And, $y=b\sin \theta$

$\Rightarrow$ $y-0=\left(4\sqrt{3}\right)\sin \theta$

$\Rightarrow$ $y=4\sqrt{3}\sin \theta$

Thus, the parametric representation of a point on the ellipse is,

$\left(x,y\right)=\left(3+8\cos \theta ,4\sqrt{3}\sin \theta \right)$

Hence, option $A$ is the correct option.