How do you convert 2y=3y^2-4x^2 -2x  into a polar equation?

Mar 27, 2017

$r = \frac{2 \left(\sin \theta + \cos \theta\right)}{3 {\sin}^{2} \theta - 4 {\cos}^{2} \theta}$

Explanation:

The relation between polar coordinates $\left(r , \theta\right)$ and Cartesian coordinates $\left(x , y\right)$ is given by $x = r \cos \theta$ and $y = r \sin \theta$

Hence, $2 y = 3 {y}^{2} - 4 {x}^{2} - 2 x$

$\Leftrightarrow 2 r \sin \theta = 3 {r}^{2} {\sin}^{2} \theta - 4 {r}^{2} {\cos}^{2} \theta - 2 r \cos \theta$

i.e. ${r}^{2} \left(3 {\sin}^{2} \theta - 4 {\cos}^{2} \theta\right) = 2 r \left(\sin \theta + \cos \theta\right)$

or $r = \frac{2 \left(\sin \theta + \cos \theta\right)}{3 {\sin}^{2} \theta - 4 {\cos}^{2} \theta}$