Question

In Problems 13–24, analyze each equation and graph it. $r(2-\cos \theta )=2$

Short answer

The equation represents an ellipse.

Step-by-step solution

Step 1. Given Information 

We are given an equation

$r(2-\cos \theta )=2$

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,

$$\begin{gathered} r=\frac{2}{2-\cos \theta } \\ r=\frac{2}{2-\cos \theta }·\frac{\frac{1}{2}}{\frac{1}{2}} \\ r=\frac{1}{1-\frac{1}{2}\cos \theta } \\ OnComparing, \\ e=\frac{1}{2}\text{and} \\ ep=1\Rightarrow \left(\frac{1}{2}\right)p=1 \\ \Rightarrow p=\frac{2}{1}\left(1\right)=2 \end{gathered}$$

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

Step 3. Plotting points 

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

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

Step 4. Required graph 

Plotting the points, we get the graph as,