Question

Convert each polar equation into a rectangular equation.

$r(3-2\sin (\theta \left)\right)=6$

Short answer

The rectangular equation is

$9x^2+5y^2-24y=36$

Step-by-step solution

Step 1. Given information

The polar equation is

$r(3-2\sin (\theta \left)\right)=6$

Step 2. Convert into the rectangular equation

The polar equation is

$r(3-2\sin (\theta \left)\right)=6$

Multiply $r$on left side

$3r-2r\sin \left(\theta \right)=6$

Apply the conversion formula

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

$3\sqrt{x^2+y^2}-2y=6$

Add both sides $2y$

$$\begin{gathered} 3\sqrt{x^2+y^2}-2y+2y=6+2y \\ 3\sqrt{x^2+y^2}=6+2y \end{gathered}$$

Take square on both sides and simplify

$$\begin{gathered} {\left(3\sqrt{x^2+y^2}\right)}^2={(6+2y)}^{2}9(x^2+y^2)=36+4y^2+24y \\ 9x^2+9y^2=36+4y^2+24y \\ 9x^2+9y^2-4y^2-24y=36 \\ 9x^2+5y^2-24y=36 \end{gathered}$$

So, the rectangular equation is

$9x^2+5y^2-24y=36$