Question

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

$r=\frac{10}{5+4\cos \theta }$

Short answer

The given equation represents a ellipse.

The graph of the equation is

Step-by-step solution

Step 1. Given Information

We are given an equation

$r=\frac{10}{5+4\cos \theta }$

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,

$$\begin{gathered} r=\frac{10}{5+4\cos \theta }·\frac{\frac{1}{5}}{\frac{1}{5}} \\ r=\frac{2}{1+\frac{4}{5}\cos \theta } \end{gathered}$$

Therefore,

$$\begin{gathered} e=\frac{4}{5}<1 \\ \left(\frac{4}{5}\right)p=2 \\ p=\frac{5}{2} \end{gathered}$$

Since,$e<1$therefore the equation represents a ellipse.

Step 3. Plotting points

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

$$\begin{gathered} \text{At}\theta =0:r=\frac{10}{5+4\cos \left(0\right)}=\frac{10}{5+4}=\frac{10}{9} \\ \text{At}\theta =\frac{\pi }{2}:r=\frac{10}{5+4\cos \left(\frac{\pi }{2}\right)}=\frac{10}{5+0}=\frac{10}{5}=2\left(2,\frac{\pi }{2}\right) \\ \text{At}\theta =\pi :r=\frac{10}{5+4\cos \left(\pi \right)}=\frac{10}{5-4}=\frac{10}{1}=10 \\ At\theta =\frac{3\pi }{2}:r=\frac{10}{5+4\cos \left(\frac{3\pi }{2}\right)}=\frac{10}{5+0}=\frac{10}{5}=2\left(2,\frac{3\pi }{2}\right) \end{gathered}$$

Step 4. Required graph

Plotting the points, we get the graph as,