find a cartesian equation for the parametrics x=(5T)/(1+t^3) and y=(9t^2)/(1+t^3)

Question

I need to find a cartesian equation for the above parametrics in the form P(x,y) = 0 and then have the coefficient of the x^3 be 729.

Answer 1

x = 5t / (1 + t³)

 

y = 9t² / (1 + t³)

 

Dividing the second equation by the first,

 

y/x = 9t/5.

 

So, t = 5y/9x.  Substitute in for t:

 

 

x = 25y/[9x(1 + 125y³/729x³)]

 

9x²(729x³ + 125y³) = 25y*729x³

 

729x³ + 125y³ = 25y*81x = 2025x

 

 

729x³ + 125y³ - 2025x = 0