Question

Convert the rectangular equation to a polar equation. Solve for r. Show all work.

Answer 1

In order to convert from rectangular to polar, we can use the following relations:

$\begin{gathered} x=r\cos (\theta) \\ y=r\sin (\theta) \end{gathered}$

So, using these values of x and y in the given equation, we have:

$\begin{gathered} x^2+5x+y^2=0 \\ r^2\cos ^2(\theta)+5r\cos (\theta)+r^2\sin ^2(\theta)=0 \\ r^2(\cos ^2(\theta)+\sin ^2(\theta))+5r\cos (\theta)=0 \\ r^2(1)+5r\cos (\theta)=0 \\ r^2+5r\cos (\theta)=0 \\ r(r+5\cos (\theta))=0 \\ \begin{cases}r=0 \\ r+5\cos (\theta)=0\to r=-5\cos (\theta)\end{cases} \end{gathered}$