Serpentine curve

A serpentine curve is a curve whose equation is of the form $x^2y+a^2y-abx=0$, where $ab > 0$. Equivalently, it has a parametric representation $x=a\cot(t)$, $y=b\sin (t)\cos(t)$, or functional representation $y=\frac{abx}{x^2+a^2}$. Serpentine curves were studied by L'Hôpital and Huygens, and named and classified by Newton.
The serpentine curve for $a=b=1.$