Question

In Problems 71–80, graph each polar equation. Verify your graph using a graphing utility.

$r=\frac{2}{1-2\cos \theta }$

Short answer

The graph of the polar equation $r=\frac{2}{1-2\cos \theta }$is:

Step-by-step solution

Step 1. Given

The polar equation: $r=\frac{2}{1-2\cos \theta }$

Step 2. The polar axis

Replace $\theta by-\theta$,

$$\begin{gathered} r=\frac{2}{1-2\cos (-\theta )} \\ =\frac{2}{1-2\cos \theta } \end{gathered}$$

The test is satisfied, so the graph is symmetric with respect to the polar axis.

Step 3. The line

Replace $\theta by\pi -\theta$,

$$\begin{gathered} r=\frac{2}{1-2\cos (\pi -\theta )} \\ =\frac{2}{1-2(\cos \pi .\cos \theta +\sin \pi \sin \theta )} \\ =\frac{2}{1+\cos \theta } \end{gathered}$$

The test fails, so the graph may or may not be symmetric with respect to the line $\theta =\frac{\mathrm{\pi }}{2}$

Step 4. The pole

Replace $rby-r$,

$$\begin{gathered} -r=\frac{2}{1-2\cos \theta } \\ r=-\frac{2}{1-2\cos \theta } \end{gathered}$$

The test fails, so the graph may or may not be symmetric with respect to the pole.

Step 5. Graph the equation

Sketch the graph of the polar equation,

Step 6. Verify the graph

Verify the graph using graphing utility.