Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$r=6\cos \theta$

Short answer

The rectangular equation is${(x-3)}^2+y^2=9$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$r=6\cos \theta$

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,$r=6\cos \theta$

Multiply by r on both sides,

$\Rightarrow r^2=6r\cos \theta$

Formulae used,

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

Substitute the values in the formulae,

$\begin{array}{l}\Rightarrow x^2+y^2=6x\\ \Rightarrow x^2-6x+y^2=0\\ \Rightarrow (x^2-6x+9)+y^2=9\\ \Rightarrow ({\left(x\right)}^2-2\times 3\times x+{\left(3\right)}^2+y^2=9\\ \because a^2-2ab+\text{Invalid <m:msup> element}{b}^2=\\ \text{Invalid <m:msup> element}\Rightarrow +y^2=9\end{array}$

It is a circle with radius of three units