Chapter 11.6, Problem 21E

a.

To determine

To show: that the conic is ellipse.

Explanation of Solution

Given information:

The polar equation is $r=\frac{4}{2-\cos \theta }$ .

Proof: since, the polar equation in standard form is $r=\frac{\left(\frac{1}{2}\right)\left(4\right)}{1-\frac{1}{2}\cos \theta }$ ,

Form the polar equation it can be observed that, $e=\frac{1}{2}\text{ and }d=4$ .

Since, $e<1$ ,

Thus, the conic is ellipse.

Now, by using plotting point method,

$\theta$	$r$
$\frac{\pi }{2}$	$2$
$3\pi$	$\frac{4}{3}$
$\pi$	$\frac{4}{3}$

Now, using the above table and join the points,

The graph can be obtained as:

  

b.

To determine

To draw: the ellipse with its directrix and vertex.

Explanation of Solution

Given information:

The polar equation is $r=\frac{4}{2-\cos \theta }$ .

Proof: since, the polar equation in standard form is $r=\frac{\left(\frac{1}{2}\right)\left(4\right)}{1-\frac{1}{2}\cos \theta }$ ,

Form the polar equation it can be observed that, $e=\frac{1}{2}\text{ and }d=4$ .

Since, $e<1$ ,

Thus, the conic is ellipse.

Now, by using plotting point method,

$\theta$	$r$
$\frac{\pi }{2}$	$2$
$3\pi$	$\frac{4}{3}$
$\pi$	$\frac{4}{3}$

Now, using the above table and join the points,

The graph can be obtained as:

  

Interpretation: from the above graph it can be observed that the vertices of the parabola is $\left(-1.33,0\right),\left(4,0\right)$ and directrix is $x=-4$ .

c.

To determine

To find : the centre and length of the major and minor axes.

Answer to Problem 21E

the centre of ellipse is $\left(\frac{4}{3},0\right)$ and the length of major axis is $\frac{16}{3}$ and length of minor axis is $\frac{8\sqrt{3}}{3}$ .

Explanation of Solution

Given information :

The polar equation is $r=\frac{4}{2-\cos \theta }$ .

Calculation : from the graph in the part (b) it can be observed that the centre of ellipse is $\left(\frac{4}{3},0\right)$ and the length of major axis is $\frac{16}{3}$ and length of minor axis is $\frac{8\sqrt{3}}{3}$ .