Question

Convert each polar equation into a rectangular equation.

$r=\frac{8}{2-\sin \left(\theta \right)}$

Short answer

The rectangular equation is

$4x^2+3y^2-16y=64$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{8}{2-\sin \left(\theta \right)}$

Step 2. Convert into the rectangular equation

The polar equation is

$r=\frac{8}{2-\sin \left(\theta \right)}$

Multiply denominator on both sides

$$\begin{gathered} r\times (2-\sin (\theta \left)\right)=\frac{8}{2-\sin \left(\theta \right)}\times (2-\sin (\theta \left)\right) \\ r(2-\sin (\theta \left)\right)=8 \\ 2r-r\sin \left(\theta \right)=8 \end{gathered}$$

Apply the conversion formula

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

$2\sqrt{x^2+y^2}-y=8$

Add both sides by $y$

$$\begin{gathered} 2\sqrt{x^2+y^2}-y+y=8+y \\ 2\sqrt{x^2+y^2}=8+y \end{gathered}$$

Take square on both sides and simplify

$$\begin{gathered} {\left(2\sqrt{x^2+y^2}\right)}^2={(8+y)}^{2}4(x^2+y^2)=64+y^2+16y \\ 4x^2+4y^2=64+y^2+16y \\ 4x^2+4y^2-y^2-16y=64 \\ 4x^2+3y^2-16y=64 \end{gathered}$$

So, the rectangular equation is

$4x^2+3y^2-16y=64$