Let's take an ellipse with the standard equation
$$\frac{x^2}{9}+\frac{y^2}{4}=1$$
And I am trying to invert the following ellipse across that ellipse
$$\frac{4x^2}{3}+4\left(y-\frac{3}{2}\right)^2=1$$
I obtain a very strange curve with the equation:
$$\frac{4\left(\frac{36x}{9y^2+4x^2}\right)^2}{3}+4\left(\left(\frac{36y}{9y^2+4x^2}\right)-\frac{3}{2}\right)^2=1$$
Does anyone know what the general equation of the type of quartic curve formed is? It looks like an egg of some sort, but I would like to know the generalized equation of that egg so that I can change its parameters and apply a horizontal shrink, for example. Thanks!