Parametrizing the folium of Descartes

[Bohrok]

After recently studying Calc III on my own, I came across this problem with parametrizing the folium of Descartes.

Homework Statement

Show that for t ≠ -1, 0, the line y = tx intersects the folium at the origin and at one other point P. Express the coordinates of P in terms of t to obtain a parametrization of the folium.

Homework Equations

The folium of Descartes is the curve with the equation x³ + y³ = 3axy, where a ≠ 0 is a constant.

The Attempt at a Solution

I couldn't really get anywhere with this problem. I understand why t ≠ -1, 0, and y = tx → t = y/x.
Using the formula for the slope between two points with the origin (0, 0) and point P(x₁, y₁) on the folium went back to t = y/x

I know that x(t) = 3at/(t³ + 1) and y(t) = 3at²/(t³ + 1) from the Internet but I couldn't find a good explanation of the steps in deriving the parametric equations. They look quite similar to the previous problem in the book which would probably help with this problem, but I couldn't do anything useful with that problem either:
Show that the line of slope t through (-1, 0) intersects the unit circle in the point with coordinates x = (1-t²)/(t² + 1), y = 2t/(t² + 1)
Conclude that these equations parametrize the unit circle with the point (-1, 0) excluded. Show further that t = y/(x + 1)

 

[tiny-tim]

Hi Bohrok!

The folium of Descartes is the curve with the equation x³ + y³ = 3axy, where a ≠ 0 is a constant.
…
I know that x(t) = 3at/(t³ + 1) and y(t) = 3at²/(t³ + 1) from the Internet but I couldn't find a good explanation of the steps in deriving the parametric equations.

Just put t = y/x into x³ + y³ = 3axy

 

[Bohrok]

Thanks so much tiny-tim! So annoying when I can't see the right path and start down a wrong one that doesn't work out...