How do you convert  x^2 + 6xy + y^2=5/2 to polar form?

Nov 10, 2016

$r = \sqrt{\frac{5}{2 \left(1 + 3 \sin \left(2 \theta\right)\right)}}$

Explanation:

Substitute ${r}^{2}$ for ${x}^{2} + {y}^{2}$

${r}^{2} + 6 x y = \frac{5}{2}$

Substitute $r \cos \left(\theta\right)$ for x and $r \sin \left(\theta\right)$ for y:

${r}^{2} + 6 {r}^{2} \cos \left(\theta\right) \sin \left(\theta\right) = \frac{5}{2}$

Factor out ${r}^{2}$

${r}^{2} \left(1 + 6 \cos \left(\theta\right) \sin \left(\theta\right)\right) = \frac{5}{2}$

Substitute $\sin \left(2 \theta\right)$ for $2 \cos \left(\theta\right) \sin \left(\theta\right)$:

${r}^{2} \left(1 + 3 \sin \left(2 \theta\right)\right) = \frac{5}{2}$

Divide both sides by $\left(1 + 3 \sin \left(2 \theta\right)\right)$:

${r}^{2} = \frac{5}{2 \left(1 + 3 \sin \left(2 \theta\right)\right)}$

$r = \sqrt{\frac{5}{2 \left(1 + 3 \sin \left(2 \theta\right)\right)}}$