Chapter 6, Problem 61RE

Solution

To find: The rectangular form of given polar equation and identify the graph.

The rectangular form of given equation is $x^2+y^2=4$ and the graph of the equation is a circle with centre $\left(0,0\right)$ and radius $2$ .

Given data:

The polar equation is $r=-2$ .

Formula used:

Let $\left(r,\theta \right)$ and $\left(x,y\right)$ are the polar and rectangular coordinates of the point P.

Then,

  $\begin{array}{c}x=r\cos \theta \\ y=r\sin \theta \\ r^2=x^2+y^2\\ \tan \theta =\frac{y}{x}\end{array}$

Calculation:

  $r=-2$

Substitute $r=\sqrt{x^2+y^2}$ .

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

Therefore, the rectangular form of given polar equation is $x^2+y^2=4$ .

Plot the graph of equation.

  

Figure (1)

From the graph, it can be observed that the graph of the equation $x^2+y^2=4$ is a circle with center $\left(0,0\right)$ and radius $2$ .

Therefore, the rectangular form of given equation is $x^2+y^2=4$ and the graph of the equation is a circle with centre $\left(0,0\right)$ and radius $2$ .