Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$r=1+\cos \theta$

Short answer

The rectangular equation is${(x^2-x+y^2)}^2=x^2+y^2$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$r=1+\cos \theta$

Step 2. Concept used

Use conversion formulae from rectangular coordinates into polar coordinates as-

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

Also use conversion formulae for polar coordinates as-

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

Substitute the value in the formulae. And simplify them.

Step 3. Calculation

The polar equation is given as,$r=1+\cos \theta$

Multiply by r on both sides,

$\begin{array}{l}\Rightarrow r^2=r(1+\cos \theta )\\ \Rightarrow r^2=r+r\cos \theta \end{array}$

Formulae used,

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

Substitute the values in the formulae,

$\begin{array}{l}\Rightarrow x^2+y^2=r+x\\ \Rightarrow x^2-x+y^2=r\\ \Rightarrow {(x^2-x+y^2)}^2=r^2\\ \Rightarrow {(x^2-x+y^2)}^2=x^2+y^2\end{array}$