# How do you convert x^2+y^2 - 2y=0 into polar form?

Jul 13, 2016

Make use of a few conversion formulas and simplify. See below.

#### Explanation:

Recall the following formulas, used for conversion between polar and rectangular coordinates:

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

Now take a look at the equation:
${x}^{2} + {y}^{2} - 2 y = 0$

Since ${x}^{2} + {y}^{2} = {r}^{2}$, we can replace the ${x}^{2} + {y}^{2}$ in our equation with ${r}^{2}$:
${x}^{2} + {y}^{2} - 2 y = 0$
$\to {r}^{2} - 2 y = 0$

Also, because $y = r \sin \theta$, we can replace the $y$ in our equation with $\sin \theta$:
${r}^{2} - 2 y = 0$
$\to {r}^{2} - 2 \left(r \sin \theta\right) = 0$

We can add $2 r \sin \theta$ to both sides:
${r}^{2} - 2 \left(r \sin \theta\right) = 0$
$\to {r}^{2} = 2 r \sin \theta$

And we can finish by dividing by $r$:
${r}^{2} = 2 r \sin \theta$
$\to r = 2 \sin \theta$