Chapter 9.2, Problem 29AYU

To determine

To find: Transform the polar equation $r = 2$ to rectangular equation. Then identify and match the graph given.

Answer to Problem 29AYU

Solution:

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

From the given graph, option E satisfies the condition of circle.

Given:

It is asked to transform the polar equation $r = 2$ to rectangular equation and match the graph.

Explanation of Solution

Convert the polar equation to a rectangular equation $= \text{}2$.

Formula used:

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

Consider $= \text{}2$,

Squaring on both the sides,

$\begin{array}{}\begin{array}{c}{r}^2 = 4\\ x^2 + y^2 = 4\\ x^2 + y^2 = 2^2\end{array}\end{array}$

The above is an equation of circle with center $\left(0, 0\right)$ and radius 2.

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

From the given graph, option E satisfies the condition of circle.