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### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

# Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates $r=\frac{2}{1-\mathrm{cos}\theta }$

The rectangular equation is${y}^{2}=4\left(1+x\right)$

See the step by step solution

## Step 1. Given information

An equation is given in polar coordinate system as $r=\frac{2}{1-\mathrm{cos}\theta }$

## Step 2. Concept used

To convert rectangular coordinates to polar coordinates, use the formulae below-

$\begin{array}{l}r=\sqrt{{x}^{2}+{y}^{2}}\\ \theta ={\mathrm{tan}}^{-1}\left(\frac{y}{x}\right)\end{array}$

Convert polar coordinates using the same formulas-

$\begin{array}{l}x=r\mathrm{cos}\theta \\ y=r\mathrm{sin}\theta \end{array}$

Substitute the value in the formulae. And simplify them.

## Step 3. Calculation

The polar equation is given as,$r=\frac{2}{1-\mathrm{cos}\theta }$

Multiply by r on both sides,

$\begin{array}{l}{r}^{2}=\frac{2r}{1-\mathrm{cos}\theta }\\ ⇒{r}^{2}=\frac{2{r}^{2}}{r-r\mathrm{cos}\theta }\\ ⇒1=\frac{2}{r-r\mathrm{cos}\theta }\\ ⇒1=\frac{2}{r-x}\\ ⇒r=x+2\\ ⇒{r}^{2}\text{Invalid element}=\\ ⇒{x}^{2}+{y}^{2}={x}^{2}+4+4x\\ ⇒{y}^{2}=4\left(1+x\right)\end{array}$