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

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