Question

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

$r=\frac{8}{2-\sin \theta }$

Short answer

The equation represents an ellipse.

Step-by-step solution

Step 1. Given Information 

We are given an equation

$r=\frac{8}{2-\sin \theta }$

We need to find the graph of the equation,

Step 2. Analyze the equation 

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

The given equation becomes,

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

Here,

$$\begin{gathered} e=\frac{1}{2} \\ ep=4\Rightarrow \left(\frac{1}{2}\right)p=4 \\ \Rightarrow p=\frac{2}{1}\left(4\right)=8 \end{gathered}$$

Since,$e<1$, therefore, it is an ellipse.

Step 3. Plotting points 

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

$$\begin{gathered} \theta =0:r=\frac{8}{2-\sin \left(0\right)}=\frac{8}{2-0}=\frac{8}{2}=4 \\ \theta =\frac{\pi }{2}:r=\frac{8}{2-\sin \left(\frac{\pi }{2}\right)}=\frac{8}{2-1}=\frac{8}{1}=8 \\ \theta =\pi :r=\frac{8}{2-\sin \left(\pi \right)}=\frac{8}{2-0}=\frac{8}{2}=4 \\ \theta =\frac{3\pi }{2}:r=\frac{8}{2-\sin \left(\frac{3\pi }{2}\right)}=\frac{8}{2+1}=\frac{8}{3}\left(\frac{8}{3},\frac{3\pi }{2}\right) \end{gathered}$$

Step 4. Required graph

Plotting the points, we get the graph as,