Can there be variations on the Witch of Agnesi?

Question

The function known as "The Witch of Agnesi" can be constructed using a circle of radius $a$, and is written in Cartesian coordinates as $$f(x)=\frac{8a^3}{x^2+4a^2}$$ The family of functions that follow this pattern are all dependent upon the circle, which gives it its "bump" shape. Can the function be modified so the circle is replaced by an ellipse with unequal major and minor axis? How would it be parameterized, and written in Cartesian coordinates?

Answer 1

Instead of a circle of radius $a$, suppose we start with an ellipse with horizontal semi-axis $a$, and vertical semi-axis $b$. Then the construction is identical, except that everything is stretched by a factor of $b/a$ in the vertical direction. So the equation is simply

$$f(x)=\frac{b}{a}\cdot\frac{8a^3}{x^2+4a^2} = \frac{8a^2b}{x^2+4a^2}$$

Answer 2

I plotted TonyK's result with Mathematica.

Here is the code.

a = 1; 
b = 0.5;
ContourPlot[{8*a^2*b/(4*a^2 + x^2) - y == 
   0, (x/a)^2 + (y/b - 1)^2 - 1 == 0}, {x, -3, 3}, {y, -3, 3}]