Question

In Problems 13–24, analyze each equation and graph it

$r=\frac{8}{2+4\cos \theta }$

Short answer

The given represents a hyperbola.

Step-by-step solution

Step 1. Given Information 

We are given an equation

$r=\frac{8}{2+4\cos \theta }$

We need to find the graph of the equation.

Step 2. Analyze the equation 

Compare the given equation with $r=\frac{ep}{1+e\cos \theta }$

The given equation becomes,

$$\begin{gathered} r=\frac{8}{2+4\cos \theta }·\frac{\frac{1}{2}}{\frac{1}{2}} \\ r=\frac{4}{1+2\cos \theta } \end{gathered}$$

Therefore,

$$\begin{gathered} e=2 \\ ep=4\Rightarrow \left(2\right)p=4 \\ \Rightarrow p=\frac{4}{2}=2 \end{gathered}$$

Since $e>1$, therefore it is a hyperbola.

Step 3. Plotting points 

Now, we will calculate the points from the given equation,

$$\begin{gathered} \theta =0:r=\frac{8}{2+4\cos \left(0\right)}=\frac{8}{2+4}=\frac{8}{6}=\frac{4}{3}\left(\frac{4}{3},0\right) \\ \theta =\frac{\pi }{2}:r=\frac{8}{2+4\cos \left(\frac{\pi }{2}\right)}=\frac{8}{2+0}=\frac{8}{2}=4\left(4,\frac{\pi }{2}\right) \\ \theta =\pi :r=\frac{8}{2+4\cos \left(\pi \right)}=\frac{8}{2-4}=\frac{8}{-2}=-4(-4,\pi ) \\ \theta =\frac{3\pi }{2}:r=\frac{8}{2+4\cos \left(\frac{3\pi }{2}\right)}=\frac{8}{2+0}=\frac{8}{2}=4 \end{gathered}$$

Step 4. Required graph

Plotting the points, we get the graph as,