Chapter 9.2, Problem 68AYU

In problems 69-72, the polar equation for each graph is either $r=a+b\cos \theta$ or $r=a+b\sin \theta ,a>0$. Select the correct equation and find the values of $a\text{ and }b$.

To determine

To find: The polar equation of the graph given which is either $r = a + b \text{cos}\theta or r = a + b \text{sin}\theta , a>0$. Conclude the values of $a$ and $b$.

Answer to Problem 68AYU

Solution The required polar equation must be $r = 3\left(1 - \text{cos}\theta \right) = 3 - 3 \text{cos}\theta$. The values of $a$ and $b$ are $a = 3 \text{and} b = - 3$.

Given:

It is asked to find the polar equations of the following graph:

Explanation of Solution

Formula to be used:

Cardioids are characterized by equations of the form

$\begin{array}{l}r = a\left(1\pm \text{cos}\theta \right)\\ r = a\left(1\pm \text{sin}\theta \right)\end{array}$

where $a>0$. The graph of a cardioid passes through the pole.

Limacons without an inner loop are characterized by equations of the form

$\begin{array}{l}r = a\pm b \text{cos}\theta \\ r = a\pm b \text{sin}\theta \end{array}$

Where $a>0, b>0 and a>b$. The graph of a Limacon without an inner loop does not pass through the pole.

Limacons with an inner loop are characterized by equations of the form

$\begin{array}{l}r = a\pm b \text{cos}\theta \\ r = a\pm b \text{sin}\theta \end{array}$

Where $a>0, b>0 and a<b$. The graph of a Limacon with an inner loop pass through the pole twice.

Given graph looks like cardioids and passes through the pole and the center point is $\left(- 3, 0\right)$

It is starts from pole $\left(0\right)$ and goes up to  − 6 in the polar axis. This cardioids lies left side of the pole. Therefore it must take the cosine angle with $a = 3$.

As Cardioids are characterized by equations of the form

$r = a\left(1\pm \text{cos}\theta \right)$

It must take $r = a\left(1 - \text{cos}\theta \right)$ as it lies left to the polar axis.

The required polar equation must be $r = 3\left(1 - \text{cos}\theta \right) = 3 - 3\text{cos}\theta$.

Hence, the values of $a$ and $b$ are $a = 3$ and $b = - 3$.