Chapter 11.6, Problem 29E

a.

To determine

To find: the eccentricity and identify the conic.

Answer to Problem 29E

The eccentricity is $3$ and the conic is hyperbola.

Explanation of Solution

Given information:

The polar equation is $r=\frac{4}{1+3\cos \theta }$ .

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

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

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

  $r=\frac{3\left(\frac{4}{3}\right)}{1+3\cos \theta }$

By comparing both polar equation,

  $e=3\text{ and }d=\frac{4}{3}$

Thus, the eccentricity of the conic is $3$ .

Because the $e>1$ ,

Thus, the conic is hyperbola.

b.

To determine

To draw: the conic.

Explanation of Solution

Given information:

The polar equation is $r=\frac{4}{1+3\cos \theta }$ .

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

$\theta$	$r$
$\frac{\pi }{2}$	$4$
$0$	$1$
$\frac{5\pi }{2}$	$4$

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 conic is $\left(1,0\right),\left(2,0\right)$ .