Differentiable curve/t^2-1,t^3-t/y^2 is x^2+x^3/Exercise

We consider the curve

$\mathbb {R} \longrightarrow \mathbb {R} ^{2},t\longmapsto (t^{2}-1,t^{3}-t).$

a) Show that the image points ${}(x,y)$ of the curve satisfy the equation

${}y^{2}=x^{2}+x^{3}\,.$

b) Show that every point ${}(x,y)\in \mathbb {R} ^{2}$ satisfying ${}y^{2}=x^{2}+x^{3}$ belongs to the image of the curve.

c) Show that there exist exactly two points ${}t_{1}$ and ${}t_{2}$ with the same image point, and that the mapping is injective after removing these points.

Create a solution