Question

Convert each polar equation into a rectangular equation.

$r(2-\cos (\theta \left)\right)=2$

Short answer

The rectangular equation is

$3x^2+4y^2-4x=4$

Step-by-step solution

Step 1. Given information

The polar equation is

$r(2-\cos (\theta \left)\right)=2$

Step 2. Convert into the rectangular equation

The polar equation is

$r(2-\cos (\theta \left)\right)=2$

Multiply left side term by $r$

$2r-r\cos \left(\theta \right)=2$

Apply the conversion formula

$r=\sqrt{x^2+y^2},x=r\cos \left(\theta \right)$

$2\sqrt{x^2+y^2}-x=2$

Add both sides by $x$

$$\begin{gathered} 2\sqrt{x^2+y^2}-x+x=2+x \\ 2\sqrt{x^2+y^2}=2+x \end{gathered}$$

Take square on both sides and simplify

$$\begin{gathered} {\left(2\sqrt{x^2+y^2}\right)}^2={(2+x)}^{2}4(x^2+y^2)=4+x^2+4x \\ 4x^2+4y^2=4+x^2+4x \\ 4x^2+4y^2-x^2-4x=4 \\ 3x^2+4y^2-4x=4 \end{gathered}$$

So, the rectangular equation is

$3x^2+4y^2-4x=4$