Chapter 11.6, Problem 23E

a.

To determine

To show: that the conic is ellipse.

Explanation of Solution

Given information:

The polar equation is $r=\frac{12}{4+3\sin \theta }$ .

Proof: since, the polar equation in standard form is $r=\frac{\left(\frac{3}{4}\right)\left(4\right)}{1+\frac{3}{4}\sin \theta }$ ,

Form the polar equation it can be observed that, $e=\frac{3}{4}\text{ and }d=4$ .

Since, $e<1$ ,

Thus, the conic is ellipse.

Now, by using plotting point method,

$\theta$	$r$
$\frac{\pi }{2}$	$\frac{12}{7}$
$0$	$3$
$\frac{5\pi }{2}$	$3$

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 $r=\frac{12}{4+3\sin \theta }$ .

Proof: since, the polar equation in standard form is $r=\frac{\left(\frac{3}{4}\right)\left(4\right)}{1+\frac{3}{4}\sin \theta }$ ,

Form the polar equation it can be observed that, $e=\frac{3}{4}\text{ and }d=4$ .

Since, $e<1$ ,

Thus, the conic is ellipse.

Now, by using plotting point method,

$\theta$	$r$
$\frac{\pi }{2}$	$\frac{12}{7}$
$0$	$3$
$\frac{5\pi }{2}$	$3$

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 $\left(0,-12\right),\left(0,1.71\right)$ and directrix is $y=4$ .

c.

To determine

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

Answer to Problem 23E

the centre of ellipse is $\left(\frac{36}{7},\frac{3\pi }{2}\right)$ and the length of major axis is $\frac{96}{7}$ and length of minor axis is $\frac{24\sqrt{7}}{7}$ .

Explanation of Solution

Given information :

The polar equation is $r=\frac{12}{4+3\sin \theta }$ .

Calculation : from the graph in the part (b) it can be observed that the centre of ellipse is $\left(\frac{36}{7},\frac{3\pi }{2}\right)$ and the length of major axis is $\frac{96}{7}$ and length of minor axis is $\frac{24\sqrt{7}}{7}$ .