# How do you convert r = 12/(4 + 8 sinx) into cartesian form?

Nov 5, 2017

${x}^{2} - 3 {y}^{2} + 12 y - 9 = 0$

#### Explanation:

As $x$ is used in Cartesian or rectangular corrdinate, I would use $\theta$ for angle in place of $x$. The relation between polar coordinates $\left(r , \theta\right)$ and Cartesian coordinates $\left(x , y\right)$ is given by

$x = r \cos \theta$ and $y = r \sin \theta$ i.e. ${r}^{2} = {x}^{2} + {y}^{2}$ and $\theta = {\tan}^{- 1} \left(\frac{y}{x}\right)$

Hence $r = \frac{12}{4 + 8 \sin \theta}$

or $4 r + 8 r \sin \theta = 12$

or $4 \sqrt{{x}^{2} + {y}^{2}} = 12 - 8 y$

or $\sqrt{{x}^{2} + {y}^{2}} = 3 - 2 y$

or ${x}^{2} + {y}^{2} = 9 + 4 {y}^{2} - 12 y$

or ${x}^{2} - 3 {y}^{2} + 12 y - 9 = 0$

which isthe equation of hyperbola.
graph{x^2-3y^2+12y-9=0 [-9.75, 10.25, -3, 7]}