How do you find the Cartesian equation from given parametric equations #x = (2t )/ (1+t^4)# and #y = (2t^3) / (1 + t^4)#?

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Answer 1 · 2017-04-21T00:10:47

We have

#xy=(4t^4)/(1+t^4)^2#

Solving for #t^4# we have

#t^4 = (2 - x y pm 2 sqrt[1 - x y])/(x y)#

or

#t = pm root(4)((2 - x y pm 2 sqrt[1 - x y])/(x y))#

now substituting into the #x# equation we have

#x y (2 - x y + 2 sqrt[1 - x y]) = x^4# which is a curve foil.

The curve is a rotated Bernoulli lemniscate.