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Q 36.

Expert-verified
Found in: Page 731

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.$r=2\mathrm{cos}\theta$

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

See the step by step solution

## Step 1. Given information.

The given equation in polar coordinates is:

$r=2\mathrm{cos}\theta$

## Step 2. Find the equation in rectangular coordinates.

$r=2\mathrm{cos}\theta$

First, multiply both sides by r,

${r}^{2}=2r\mathrm{cos}\theta \phantom{\rule{0ex}{0ex}}{r}^{2}=2x\left[\mathrm{Since}r\mathrm{cos}\theta =x\right]\phantom{\rule{0ex}{0ex}}{x}^{2}+{y}^{2}=2x\left[Since{r}^{2}={x}^{2}+{y}^{2}\right]$

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

role="math" localid="1649264450788" ${x}^{2}+{y}^{2}-2x=2x-2x\phantom{\rule{0ex}{0ex}}{x}^{2}+{y}^{2}-2x=0$

Complete the square in x,

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

now add 1 on both sides of the equation,

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

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