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## 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

*x*

^{3}+

*y*

^{3}= 3

*axy*, 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(x

_{1}, y

_{1}) on the folium went back to t = y/x

I know that x(t) = 3at/(t

^{3}+ 1) and y(t) = 3at

^{2}/(t

^{3}+ 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-t

^{2})/(t

^{2}+ 1), y = 2t/(t

^{2}+ 1)

Conclude that these equations parametrize the unit circle with the point (-1, 0) excluded. Show further that t = y/(x + 1)