# How do you convert 3y= 2x^2-2xy-x  into a polar equation?

Oct 5, 2016

$\frac{3 \sin \left(\theta\right) + \cos \left(\theta\right)}{2 {\cos}^{2} \left(\theta\right) - 2 \cos \left(\theta\right) \sin \left(\theta\right)} = r$

#### Explanation:

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

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

Make the necessary substitutions

$3 r \sin \left(\theta\right) = 2 {\left(r \cos \left(\theta\right)\right)}^{2} - 2 r \cos \left(\theta\right) r \sin \left(\theta\right) - r \cos \left(\theta\right)$

Simplify

$3 r \sin \left(\theta\right) = 2 {r}^{2} {\cos}^{2} \left(\theta\right) - 2 {r}^{2} \cos \left(\theta\right) \sin \left(\theta\right) - r \cos \left(\theta\right)$

Add $r \cos \left(\theta\right)$ to both sides

$3 r \sin \left(\theta\right) + r \cos \left(\theta\right) = 2 {r}^{2} {\cos}^{2} \left(\theta\right) - 2 {r}^{2} \cos \left(\theta\right) \sin \left(\theta\right)$

Factor out $r$ and ${r}^{2}$

$r \left(3 \sin \left(\theta\right) + \cos \left(\theta\right)\right) = {r}^{2} \left(2 {\cos}^{2} \left(\theta\right) - 2 \cos \left(\theta\right) \sin \left(\theta\right)\right)$

Isolated ${r}^{2}$

$\frac{r \left(3 \sin \left(\theta\right) + \cos \left(\theta\right)\right)}{2 {\cos}^{2} \left(\theta\right) - 2 \cos \left(\theta\right) \sin \left(\theta\right)} = \frac{{r}^{2} \cancel{2 {\cos}^{2} \left(\theta\right) - 2 \cos \left(\theta\right) \sin \left(\theta\right)}}{\cancel{2 {\cos}^{2} \left(\theta\right) - 2 \cos \left(\theta\right) \sin \left(\theta\right)}}$

$\frac{r \left(3 \sin \left(\theta\right) + \cos \left(\theta\right)\right)}{2 {\cos}^{2} \left(\theta\right) - 2 \cos \left(\theta\right) \sin \left(\theta\right)} = {r}^{2}$

Gather $r$ to the right hand side

$\frac{\frac{\cancel{r} \left(3 \sin \left(\theta\right) + \cos \left(\theta\right)\right)}{2 {\cos}^{2} \left(\theta\right) - 2 \cos \left(\theta\right) \sin \left(\theta\right)}}{\cancel{r}} = {r}^{\cancel{2}} / \cancel{r}$

Simplify

$\frac{3 \sin \left(\theta\right) + \cos \left(\theta\right)}{2 {\cos}^{2} \left(\theta\right) - 2 \cos \left(\theta\right) \sin \left(\theta\right)} = r$

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