Eliminate the parameter x = t^2 - 1 and
y = t^2 + t
t^2 = x+1
y = x + 1 +/- (x+1)^.5
you can manipulate that further
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
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)
To eliminate the parameter, we need to express one variable in terms of the other variable. In this case, we want to express y in terms of x.
Given:
x = t^2 - 1
y = t^2 + t
Let's solve the first equation for t:
x = t^2 - 1
Add 1 to both sides:
x + 1 = t^2
Take the square root of both sides (considering both positive and negative roots):
√(x + 1) = ±t
Now, substitute t = √(x + 1) into the second equation:
y = (√(x + 1))^2 + √(x + 1)
Simplifying:
y = x + 1 + √(x + 1)
Therefore, by eliminating the parameter t, we express y in terms of x as:
y = x + 1 + √(x + 1)