Question

Convert each polar equation into a rectangular equation.

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

Short answer

The rectangular equation is

$x^2-6y=9$

Step-by-step solution

Step 1. Given information

The polar equation is

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

Step 2. Convert into the rectangular equation

The polar equation is

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

Multiply top and bottom by $\sin \left(\theta \right)$

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

Multiply denominator on both sides

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

Apply the conversion formula

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

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

Add both sides by $y$

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

Take square on both sides and simplify it

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

So, the rectangular equation is

$x^2-6y=9$