Chapter 9.2, Problem 70AYU

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 70AYU

Solution

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

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 $a<\mathrm{b}$. The graph of a Limacon with an inner loop pass through the pole twice.

Given graph looks like Limacon with an inner loop and the graph passes through the pole twice.

It touches the polar axis upto at (1, 0). This Limacon lies upside of the pole. It must take the sine angle with $a=1$, $b=5$ as it till 5 in the line.

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

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

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

The required polar equation must be $r=1+4\text{sin}\theta$.

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