Chapter 11.6, Problem 40E

a.

To determine

To find: the eccentricity and directrix of the conic.

Answer to Problem 40E

The eccentricity is $1$ and the directrix of the conic is $x=\frac{9}{2}$ .

Explanation of Solution

Given information:

The polar equation is $r=\frac{9}{2+2\cos \theta },\theta =-\frac{5\pi }{6}$ .

Calculation: to find the eccentricity and directrix, first compare the polar equation with the standard equation,

The standard equation is $r=\frac{ed}{1+e\sin \theta }$ .

Now, convert the polar equation in form of standard form of equation,

  $r=\frac{1\left(\frac{9}{2}\right)}{1+\cos \theta }$

By comparing both polar equation,

  $e=1\text{ and }d=\frac{9}{2}$

Thus, the eccentricity of the conic is $1$ .

And the directrix of the conic is $x=\frac{9}{2}$ .

b.

To determine

To find: the polar equation after rotating the conic.

Answer to Problem 40E

The polar equation of conic is $r=\frac{9}{2+2\cos \left(\theta +\frac{5\pi }{6}\right)}$ after rotating.

Explanation of Solution

Given information:

The polar equation is $r=\frac{9}{2+2\cos \theta },\theta =-\frac{5\pi }{6}$ .

Calculation: to find the polar equation, substitute $\theta =\theta +\frac{5\pi }{6}$ in the equation,

  $r=\frac{9}{2+2\cos \left(\theta +\frac{5\pi }{6}\right)}\text{ [substitute]}$

Thus, the polar equation is $r=\frac{9}{2+2\cos \left(\theta +\frac{5\pi }{6}\right)}$ .

c.

To determine

To draw: the original conic and after rotating.

Explanation of Solution

Given information:

The polar equation is $r=\frac{9}{2+2\cos \theta },\theta =-\frac{5\pi }{6}$ .

Graph: to draw the conic, use the plotting point method,

$\theta$	$r$
$0$	$3$
$\frac{\pi }{2}$	$\frac{9}{2}$
$2\pi$	$3$

By using the above table, and the graph of the conic after rotating can be obtained as:

  

Interpretation: from the above graph it can be observed that the rotating graph can be obtained by rotate the original graph $-\frac{5\pi }{6}$ .