Chapter 9.2, Problem 64AYU

In Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.

$r=3;r=2+2\cos \theta$

To determine

To find: Graph the polar equations $r = 3; r = 2 + 2 \text{cos} \theta$ on the same polar grid and find the point of intersections if any.

Answer to Problem 64AYU

Solution

The graph of the polar equations $r = 3; r = 2 + 2 \text{cos} \theta$ are equation of an circle and equation of a cardioids.

From the above graph, the point of intersections are $\left(1.5, 2.6\right) \text{and} \left(1.5, -2.6\right)$.

Given:

It is asked to graph the polar equations $r = 3; r = 2 + 2 \text{cos} \theta$.

Explanation of Solution

Formula to be used:

$\left(r \text{cos} \theta , r \text{sin} \theta \right) = \left(x, y\right)$

$x^2 + y^2=a^2$

$a = \sqrt{x^2 + y^2}, a > 0$

Equation of circle with centre at the pole, radius $a$ is given by

$r = a, a > 0$

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.

Consider $r = 3$,

It is polar equation of circle with center at pole and radius 3.

Consider $r = 2 + 2 \text{cos} \theta$,

This polar equation can be written as

$r = 2\left(1 + \text{cos} \theta \right)$

The above equation is an equation of cardioids passes through the pole.

To sketch:

$r = 3; r = 2 + 2 \text{cos} \theta$

From the above graph, the point of intersections are $\left(1.5, 2.6\right) \text{and} \left(1.5, -2.6\right)$.