# How do I find the polar equation for x^2+y^2=7y?

Sep 21, 2014

To solve this problem we need to convert all of the $x$'s and $y$'s to $r$'s and $\theta$'s.

Before we look at the specifics of this problem please take a look at the relationships between the polar and rectangular coordinate systems.

Understanding these relationships are critical to better understanding how to solve these types of problems.

$x = r \cos \left(\theta\right)$

$y = r \sin \left(\theta\right)$

$\frac{y}{x} = \tan \left(\theta\right)$

${\tan}^{1} \left(\frac{y}{x}\right) = \theta$

${x}^{2} + {y}^{2} = {r}^{2}$

To solve this type of problem one of the first things to look for is a way to make substitutions.

${x}^{2} + {y}^{2} = 7 y$

Substitute in ${r}^{2}$ for ${x}^{2} + {y}^{2}$

${r}^{2} = 7 y$

Substitute in $r \sin \left(\theta\right)$ for $y$

${r}^{2} = 7 r \sin \left(\theta\right)$

$\frac{{r}^{2}}{r} = \frac{7 r \sin \left(\theta\right)}{r}$

$r = 7 \sin \left(\theta\right) \to$ is the polar form of $\to {x}^{2} + {y}^{2} = 7 y$