# How do you convert 5y=(x+y)^2 -2xy  into a polar equation?

Mar 15, 2018

$r = 5 \sin \theta$

#### Explanation:

The relationn between Cartesian or rectangular coordinates $\left(x , y\right)$ and polar coordinates $\left(r , \theta\right)$ is given by

$x = r \cos \theta$, $y = r \sin \theta$ and hence ${x}^{2} + {y}^{2} = {r}^{2}$

Therefore we can write $5 y = {\left(x + y\right)}^{2} - 2 x y$ as

$5 r \sin \theta = {x}^{2} + {y}^{2} + 2 x y - 2 x y$

or $5 r \sin \theta = {x}^{2} + {y}^{2}$

or $5 r \sin \theta = {r}^{2}$

or $r = 5 \sin \theta$

This is the equation of a circle

graph{5y=(x+y)^2-2xy [-10.13, 9.87, -2.92, 7.08]}