Question

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=2\cos \theta$

Short answer

The required equation is${\left(x-1\right)}^2+y^2=1$.

Step-by-step solution

Step 1. Given information.

The given equation in polar coordinates is:

$r=2\cos \theta$

Step 2. Find the equation in rectangular coordinates.

$r=2\cos \theta$

First, multiply both sides byr,

$$\begin{gathered} r^2=2r\cos \theta \\ {r}^2=2x\left[\mathrm{Since}r\cos \theta =x\right] \\ {x}^2+y^2=2x\left[Since{r}^2=x^2+y^2\right] \end{gathered}$$

Now add $-2x$on both sides of the equation,

role="math" localid="1649264450788" $$\begin{gathered} x^2+y^2-2x=2x-2x \\ {x}^2+y^2-2x=0 \end{gathered}$$

Complete the square in x,

${\left(x-1\right)}^2+y^2-1=0$

now add 1 on both sides of the equation,

$$\begin{gathered} {\left(x-1\right)}^2+y^2-1+1=0+1 \\ {\left(x-1\right)}^2+y^2=1 \end{gathered}$$

Therefore, the equation in rectangular coordinates is${\left(x-1\right)}^2+y^2=1$.