# Equation of Ovals of Cassini/Cartesian Form

## Theorem

Let $P_1$ and $P_2$ be points in the cartesian coordinate plane located at $\tuple {a, 0}$ and $\tuple {-a, 0}$ for some constant $a \ne 0$.

Let $b$ be a real constant.

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

describes the ovals of Cassini.

## Proof

The ovals of Cassini are the loci of points $M$ in the plane such that:

$P_1 M \times P_2 M = b^2$

Let $b$ be chosen.

Let $P = \tuple {x, y}$ be an arbitrary point of $M$.

We have:

 $\displaystyle P_1 P$ $=$ $\displaystyle \sqrt {\paren {x - a}^2 + y^2}$ Distance Formula $\displaystyle P_2 P$ $=$ $\displaystyle \sqrt {\paren {x + a}^2 + y^2}$ Distance Formula $\displaystyle \leadsto \ \$ $\displaystyle \paren {P_1 P} \paren {P_2 P}$ $=$ $\displaystyle \sqrt {\paren {x + a}^2 + y^2} \sqrt {\paren {x - a}^2 + y^2}$ Definition of Ovals of Cassini $\displaystyle$ $=$ $\displaystyle b^2$ $\displaystyle \leadsto \ \$ $\displaystyle b^4$ $=$ $\displaystyle \paren {\paren {x + a}^2 + y^2} \paren {\paren {x - a}^2 + y^2}$ squaring both sides $\displaystyle$ $=$ $\displaystyle \paren {x^2 + y^2 + a^2 + 2 a x} \paren {x^2 + y^2 + a^2 - 2 a x}$ multiplying out and rearranging $\displaystyle$ $=$ $\displaystyle \paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2$ Difference of Two Squares

Hence the result.

$\blacksquare$

## Source of Name

This entry was named for Giovanni Domenico Cassini.