Chapter 11.3, Problem 28E

To determine

To determine: The equivalent Cartesian equation for the given polar equation. Also, identify the graph without using a grapher.

Answer to Problem 28E

The required equation is ${\left(x+2\right)}^2+y^2=4$ and the equation is an equation of a circle and its radius is 2 and center is $\left(-2,0\right)$ .

Explanation of Solution

Given information:

The polar equation: $r^2=-4r\cos \theta$

Calculation:

The given equation is $r^2=-4r\cos \theta$ .

To find the required Cartesian equation, use the formulas $x=r\cos \theta$ , $y=r\sin \theta$ , $r^2=x^2+y^2$ , $\sin 2\theta =2\sin \theta \cos \theta$ and $\tan \theta =\frac{y}{x}$ .

Substitute $r\cos \theta$ for $x$ and $x^2+y^2$ for $r^2$ in the equation $r^2=-4r\cos \theta$ .

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

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

Hence, the required equation is ${\left(x+2\right)}^2+y^2=4$ and the equation is an equation of a circle and its radius is 2 and center is $\left(-2,0\right)$ .