# Name of the family of curves

I tried to generalize the Lemniscate of Gerono.

Instead of $$x = dfrac{t^{2} – 1}{t^{2}+1}$$ and $$y = dfrac{2t(t^{2}-1)}{(t^{2} – 1)^{2}}$$, I used $$x = dfrac{a(t^{2} – 1)}{t^{2}+1}$$ and $$y = dfrac{bt(t^{2}-1)}{(t^{2} – 1)^{2}}$$.

When eliminating $$t$$, I used the positive root instead of both. What I got is this:

$$y = dfrac{bx}{a}left(dfrac{a – x}{2a}right)sqrt{dfrac{a + x}{a – x}}$$

Out of curiosity, I replaced $$y$$ by $$y^{2}$$. Can I ask what is the name of the family of the curve
$$y^{2} = dfrac{bx}{a}left(dfrac{a – x}{2a}right)sqrt{dfrac{a + x}{a – x}}$$?