Eliminate the parameter x = t^2 - 1 and

Question

y = t^2 + t

Answer 1

t^2 = x+1

y = x + 1 +/- (x+1)^.5
you can manipulate that further

Answer 2

x^2 = t^4-2t^2+1
y^2 = t^4+2t^3+t^2
xy = t^4+t^3-t^2-1

x^2-2xy+y^2 = -3t^2 = -3(x+1)
x^2-2xy+y^2+3x+3 = 0

Answer 3

oops
y-x = t+1
(y-x)^2 = t^2+2t+1
= 1/2(t^2+t) + 1/2 t^2 + 1
= y/2 + 1/2(t^2-1) + 3/2
= y/2 + x/2 + 3/2
(x-y)^2 = 1/2 (x+y+3)