Chapter 9.2, Problem 21AYU

In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation.

$r=2\cos \theta$

To determine

To find: Transform the polar equation $r = 2 \text{cos}\theta$ to rectangular equation. Then identify and graph the equation.

Answer to Problem 21AYU

Solution

The graph of $r = 2 \text{cos}\theta$ is an equation of circle ${\left(x - 1\right)}^2 + {\text{y}}^2 = 1^2$ with center (1, 0) and radius 1.

Given:

It is asked to transform the polar equation $r = 2 \text{cos}\theta$ to rectangular equation and graph it.

Explanation of Solution

Convert the polar equation to a rectangular equation $r = 2 \text{cos}\theta$

Formula used:

$\begin{array}{c}\left(x, y\right) = \left(r \text{cos}\theta , r \text{sin}\theta \right)\\ x^2 + y^2 = {\text{r}}^2\end{array}$

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

Multiply both the sides by $r$,

$\begin{array}{c}{r}^2 = 2r \text{cos}\theta \\ x^2 + y^2 = 2x\\ x^2 - 2\text{x} + y^2 = 0\end{array}$

Now use completion of square for ‘$y$’.

$\begin{array}{c}{x}^2 - 2\text{x} + 1 + y^2 = 1\\ {\left(x - 1\right)}^2 + {\text{y}}^2 = 1^2\end{array}$

The graph of $r = 2 \text{cos}\theta$ is an equation of circle ${\left(x - 1\right)}^2 + {\text{y}}^2 = 1^2$ with center (1,0) and radius 1.

To sketch:

$r = 2 \text{cos}\theta$