Convert $r=1+2\cos(2\theta)$ to Cartesian

Question

I want to convert $r=1+2\cos 2\theta$ to Cartesian.

$r=1+2(\cos^2\theta -\sin^2\theta)$

$r=1+2\left(\dfrac{x^2}{r^2}-\dfrac{y^2}{r^2}\right) \iff \dfrac{r-1}{2}=\dfrac{x^2-y^2}{x^2+y^2}$

$r$ won't go away.

$(r-1)^2=r^2-2r+1$

No matter how I do it, the $r$ stays. What am I supposed to do?

Answer 1

$x = r \cos \theta, y = r \sin \theta$

Given curve $\, r = 1 + 2 \cos 2\theta = 4 \cos^2 \theta - 1$

$\implies \sqrt{x^2+y^2} = \frac{4x^2}{x^2+y^2} - 1$

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

$\implies (x^2+y^2)^3 = (3x^2 - y^2)^2$