Chapter 10.9, Problem 30E

To determine

To graph: The polar equation $r=\frac{10}{1+\cos \theta }$ with help of graphing utility and identify the conic.

Explanation of Solution

Given information:

The polar equation $r=\frac{10}{1+\cos \theta }$ .

Graph:

Steps to plot the graph of the polar equation $r=\frac{10}{1+\cos \theta }$ with help of graphing utility are as follows:

Step 1: Press $\boxed{\text{MODE}}$ key.

Step 2: Use the down arrow key to reach $\boxed{\text{FUNC}}$ option.

Step 3: Use the right arrow key to reach the $\boxed{\text{Pol}}$ option.

Step 4: Press $\boxed{\text{ENTER}}$ key.

Step 5: Press $\boxed{\text{Y}=}$ key.

Step 6: Enter the function $10\boxed{÷}\boxed{(}1+\boxed{\text{COS}}\boxed{\left(\theta \right)}\boxed{)}$ .

Step 7: Press $\boxed{\text{WINDOW}}$ key. Change the settings to

  $\begin{array}{c}X\min =-10\\ X\max =10\\ Y\min =-10\\ Y\max =10\end{array}$

For better view of the graph.

Step 8: Press $\boxed{\text{GRAPH}}$ key.

The result obtained on the screen is provided below,

  

Interpretation:

When polar equations are either of the form $r=\frac{ep}{1\pm e\sin \theta }$ or $r=\frac{ep}{1\pm e\cos \theta }$ then it is called a conic. Here $e>0$ is the eccentricity and $\left|p\right|$ is the distance between the focus also called pole and the directrix.

When $0<e<1$ , conic is called ellipse.

When $e=1$ , conic is called parabola.

When $e>1$ , conic is called hyperbola.

For $r=\frac{10}{1+\cos \theta }$ , rewrite the equation as, $r=\frac{10\left(1\right)}{1+\cos \theta }$ so eccentricity is $e=1$ .

Thus, the conic represented by polar equation $r=\frac{10}{1+\cos \theta }$ is a parabola and can also be observed from graph.