A curve is given by parametric equations x=t^2−1,y=t+1. Eliminate the parameter t to find a Cartesian equation of the curveSimplify your answer
t = y-1, so
x = (y-1)^2 - 1
That's a parabola with vertex at (-1,1) opening to the right.
To eliminate the parameter t and find a Cartesian equation of the curve, we can solve the given set of parametric equations for t and substitute it back into the other equation.
Given: x = t^2 - 1 (equation 1)
y = t + 1 (equation 2)
From equation 2, we can solve for t:
t = y - 1
Now, substitute this value of t into equation 1:
x = (y - 1)^2 - 1
Expanding and simplifying equation 3:
x = y^2 - 2y + 1 - 1
x = y^2 - 2y
Therefore, the Cartesian equation of the curve is x = y^2 - 2y.
To simplify the answer:
We can rewrite the equation as x = y(y - 2).
So, the simplified Cartesian equation of the curve is x = y(y - 2).