Chapter 9.2, Problem 67AYU

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 $b$.

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 $a = 3 \text{and} b = 3$.

Explanation of Solution

Given:

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

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 (3, 0)

It is starts from pole (0) and goes up to 6 in the polar axis. This cardioids lies right 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 right 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 $b = 3$.