I am trying to plot the curve described by the equality $y^2(a+x)=x^2(a-x)$. It is clear that the domain of the expression is $x\in (-a,a]$. This requires a simple application of the general methods of curve tracing for the graph of $y\sqrt{a+x}=x\sqrt{a-x}$. And then the obtained graph can be mirrored in the $x$-axis due to symmetry.
I am trying to do this using a different method, that I suppose is called polar curve tracing. Substituting $x=a\cos(2\theta)$ seems favorable due to two reasons, one, the domain of the function is satisfied by $x=a\cos(2\theta)\forall\theta\in\mathbb{R}-\{\theta:\cos(2\theta)+1=0\}$ and, two, the rational expression $\frac{a-x}{a+x}$ simplifies to $\tan^2(\theta)$. And so the curve becomes $y^2=a^2\cos^{2}(2\theta)\tan^2\theta$. Or for what it's worth $y=\pm a\cos(2\theta)\tan(\theta)$.
But I don't see how to map the curve from the $\theta y$ plane to the $xy$ plane. Any hints are appreciated. Thanks.