Chapter 6.4, Problem 38E

Solution

To convert: The polar equation to rectangular forms and identifies the graph, also support the answer by graphing the polar equation

The rectangular form of equation is ${\left(x+2\right)}^2+y^2=4$ and a circle with center at $\left(-2,0\right)$ and radius $2$ .

Given data:

The equation is $r=-4\cos \theta$ .

Formula used:

Use the following conversion relationships between a polar coordinate $\left(r,\theta \right)$ and its corresponding rectangular coordinate $\left(x,y\right)$ .

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

Calculation:

Multiply both sides by $r$ .

  $r^2=-4r\cos \theta$

Substitute $r^2=x^2+y^2$ and $x=r\cos \theta$ in the above equation.

  $\begin{array}{c}{x}^2+y^2=-4x\\ x^2+4x+y^2=0\\ \left(x^2+4x+{\left(\frac{4}{2}\right)}^2\right)+y^2=0+{\left(\frac{4}{2}\right)}^2\\ {\left(x+2\right)}^2+y^2=4\end{array}$

The graph is a circle with center at $\left(-2,0\right)$ and radius $2$ .

This is supported by using a graphing calculator in Polar and Radian mode as shown below.

  

Therefore, the rectangular form of equation is ${\left(x+2\right)}^2+y^2=4$ and a circle with center at $\left(-2,0\right)$ and radius $2$ .