How do you find an equivalent equation of x^2 + 4y^2 = 4 in polar coordinates?

Apr 11, 2018

${r}^{2} = \frac{4}{{\cos}^{2} \theta + 4 {\sin}^{2} \theta}$

$r = \sqrt{\frac{4}{{\cos}^{2} \theta + 4 {\sin}^{2} \theta}} = \frac{2}{\sqrt{{\cos}^{2} \theta + 4 {\sin}^{2} \theta}}$

Explanation:

We'll use the two formulae:
$x = r \cos \theta$
$y = r \sin \theta$

${x}^{2} = {r}^{2} {\cos}^{2} \theta$
${y}^{2} = {r}^{2} {\sin}^{2} \theta$

${r}^{2} {\cos}^{2} \theta + 4 {r}^{2} {\sin}^{2} \theta = 4$

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

${r}^{2} = \frac{4}{{\cos}^{2} \theta + 4 {\sin}^{2} \theta}$

$r = \sqrt{\frac{4}{{\cos}^{2} \theta + 4 {\sin}^{2} \theta}} = \frac{2}{\sqrt{{\cos}^{2} \theta + 4 {\sin}^{2} \theta}}$