# Which conic section has the polar equation r=2/(3-cosq)?

Aug 18, 2016

$8 {x}^{2} + 9 {y}^{2} - 4 x - 4 = 0$

#### Explanation:

From $r = \frac{2}{3 - \cos q} \to 3 r - r \cos q = 2$

but $r \cos q = x$ and ${r}^{2} = {x}^{2} + {y}^{2}$

so

$3 r - x = 2 \to r = \frac{x + 2}{3}$ and also

${r}^{2} = {x}^{2} + {y}^{2} = {\left(x + 2\right)}^{2} / 9$

After some simplifications

$8 {x}^{2} + 9 {y}^{2} - 4 x - 4 = 0$

which is the equation of an ellipse