Question

Which of the following represents the rectangular equation x2 + y2 − 10y = 0 in a polar equation? r = 10sin θ r = 10cos θ r = 10 r = 10tan θ

Answer 1

We will have the following:

First, we remember that:

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

So:

$x^2+y^2-10y=0\Rightarrow(r\cos (\theta))^2+(r\sin (\theta))^2-10(r\sin (\theta))=0$$\Rightarrow r^2\sin ^2(\theta)+r^2\cos ^2(\theta)-10r\sin (\theta)=0\Rightarrow r^2(\sin ^2(\theta)+\cos ^2(\theta))-10r\sin (\theta)=0$$\Rightarrow r^2(1)-10r\sin (\theta)=0\Rightarrow r(r-10\sin (\theta))=0$$\Rightarrow r-10\sin (\theta)\Rightarrow r=10\sin (\theta)$