Equivalence of Definitions of Folium of Descartes

Theorem

The following definitions of the concept of Folium of Descartes are equivalent:

The folium of Descartes is the locus of the equation:

$x^3 + y^3 - 3 a x y = 0$

$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$

Then by definition:

$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$

Hence:

$\ds x^3 + y^3$

$=$

$\ds \dfrac {\paren {3 a t}^3} {\paren {1 + t^3}^3} + \dfrac {\paren {3 a t^2}^3} {\paren {1 + t^3}^3}$

$\ds $

$=$

$\ds \dfrac {\paren {3 a t}^3 \paren {1 + t^3} } {\paren {1 + t^3}^3}$

$\ds $

$=$

$\ds \dfrac {3 a \paren {3 a t} \paren {3 a t^2} } {\paren {1 + t^3}^2}$

$\ds $

$=$

$\ds 3 a \dfrac {3 a t} {1 + t^3} \dfrac {3 a t^2} {1 + t^3}$

$\ds $

$=$

$\ds 3 a x y$

$\blacksquare$