Question

Convert each polar equation into a rectangular equation.

$r=\frac{3}{1-\sin \left(\theta \right)}$

Short answer

The rectangular equation is

$x^2-6y=9$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{3}{1-\sin \left(\theta \right)}$

Step 2. Convert into the polar equation

The polar equation is

$r=\frac{3}{1-\sin \left(\theta \right)}$

Multiply denominator on both sides

$$\begin{gathered} r\times (1-\sin (\theta \left)\right)=\frac{3}{(1-\sin (\theta \left)\right)}\times (1-\sin (\theta \left)\right) \\ r\times (1-\sin (\theta \left)\right)=3 \\ r-r\sin \left(\theta \right)=3 \end{gathered}$$

Apply conversion equation formula

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

So, the rectangular equation is

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

Step 3. Remove square root

The polar equation is

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

Add both sides $y$

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

Take square on both sides

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

So, the rectangular equation is

$x^2-6y=9$