Chapter 9.2, Problem 69AYU

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 69AYU

Solution

The required polar equation must be $r=4+\text{sin} \theta$. The values of $a$ and $b$ are $a=4 \text{and} b=1$.

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

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

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

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

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

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

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

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

Given graph looks like Limacon without an inner loop and the graph does not passes through the pole and the center point is $\left(0, 1\right)$

It touches the polar axis at $\left(4, 0\right)$. This Limacon mostly lies upside of the pole. It must take the sine angle with $a=4$, $b=1$.

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

$r=a\pm b \text{sin}\theta$

It must take $r=a+b \text{cos}\theta$ as it lies upside of the pole.

The required polar equation must be $r=4+\text{sin}\theta$.

Hence, the values of $a$ and $b$ are $a=4$ and $b=1$.