Question

In Problems5–7, convert the polar equation to a rectangular equation. Graph the equation.

$r{\sin }^2\theta +8\sin \theta =r$

Short answer

The rectangular equation is $8y=x^2$.

The graph is as follows,

Step-by-step solution

Step 1. Given information  

We are given a polar equation,

$r{\sin }^2\theta +8\sin \theta =r$

We have to convert the polar equation to a rectangular equation.

Step 2. Converting the polar equation to rectangular equation 

We know that,

$$\begin{gathered} x=r\cos \theta \text{and} \\ y=r\sin \theta \\ \sin \theta =\frac{y}{r} \\ {x}^2+y^2=r^2 \end{gathered}$$

So,

$$\begin{gathered} r{\sin }^2\theta +8\sin \theta =r \\ r·{\left(\frac{y}{r}\right)}^2+8\times \frac{y}{r}=r \\ \frac{y^2}{r}+\frac{8y}{r}=r \\ {y}^2+8y=r^2 \\ 8y=x^2 \end{gathered}$$

Hence, the rectangular equation is $8y=x^2$.

Step 3. Graphing the equation 

The graph is as follows,