Question

Convert each polar equation into a rectangular equation.

$r=\frac{8}{4+3\sin \left(\theta \right)}$

Short answer

The rectangular equation is

$16x^2+7y^2+48y=64$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{8}{4+3\sin \left(\theta \right)}$

Step 2. Convert into the polar equation

The polar equation is

$r=\frac{8}{4+3\sin \left(\theta \right)}$

Multiply both sides by the denominator

$$\begin{gathered} r\times (4+3\sin (\theta \left)\right)=\frac{8}{4+3\sin \left(\theta \right)}\times 4+3\sin \left(\theta \right) \\ r\times (4+3\sin (\theta \left)\right)=8 \\ 4r+3r\sin \left(\theta \right)=8 \end{gathered}$$

Apply conversion equation formula

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

So, the rectangular equation is

$4\sqrt{x^2+y^2}+3y=8$

Step 3. Remove square root

The rectangular equation is

$4\sqrt{x^2+y^2}+3y=8$

Subtract both sides by $3y$

$$\begin{gathered} 4\sqrt{x^2+y^2}+3y-3y=8-3y \\ 4\sqrt{x^2+y^2}=8-3y \end{gathered}$$

Take square on both sides

$$\begin{gathered} {\left(4\sqrt{x^2+y^2}\right)}^2={(8-3y)}^{2}16(x^2+y^2)=64-48y+9y^2 \\ 16x^2+16y^2+48y-9y^2=64 \\ 16x^2+7y^2+48y=64 \end{gathered}$$

So, the rectangular equation is

$16x^2+7y^2+48y=64$