Equation of Witch of Agnesi/Cartesian

Theorem

The equation of the Witch of Agnesi is given in cartesian coordinates as:

$y = \dfrac {8 a^3} {x^2 + 4 a^2}$

Let $P = \tuple {x, y}$ and $A = \tuple {d, y}$.

We have that:

$\dfrac {OM} {MN} = \dfrac {2 a} x = \dfrac y d$.

$\paren {a - y}^2 + d^2 = a^2 \implies y \paren {2 a - y} = d^2$

Eliminating $d$ gives us:

$\dfrac {y^2} {y \paren {2 a - y} } = \dfrac {\paren {2 a}^2} {x^2}$

Hence:

$\dfrac y {2 a - y} = \dfrac {4 a^2} {x^2}$

from which:

$y = \dfrac {8 a^3} {x^2 + 4 a^2}$

$\blacksquare$

The cartesian equation of the Witch of Agnesi can also be seen presented as:

$x^2 y = 4 a^2 \paren {2 a - y}$

Some sources present the curve with the $y$-axis as the asymptote:

$x y^2 = 4 a^2 \paren {2 a - x}$