Equation of Witch of Agnesi/Cartesian

Theorem

 * WitchOfAgnesi.png

The equation of the Witch of Agnesi is given in cartesian coordinates as:
 * $y = \dfrac {8 a^3} {x^2 + 4 a^2}$

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

We have that:
 * $\dfrac {OM} {MN} = \dfrac {2 a} x = \dfrac y d$.

Also, by Pythagoras's Theorem:
 * $\paren {a - y}^2 + d^2 = a^2 \implies y \paren {2 a - y} = d^2$

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