1. Cubics

(2.1) Examples of parametrised cubics. Some plane cubic curves can be parametrised, just as the conics:

Nodal cubic $C: (y^2=x^3+x^2) \subset \mathbb{R}^2$ is the image of the map $\varphi: \mathbb{R}^1 \to \mathbb{R}^2$ given by $t\mapsto (t^2-1, t^3-t)$

Cuspidal cubic. $C: (y^2 = x^3)\subset \mathbb{R}^2$ is the image of $\varphi:\mathbb{R}^1 \to \mathbb{R}^2$ given by $t\mapsto (t^2,t^3)$

(1) Let $C: (y^2 = x^3 + x^2) \subset \mathbb{R}^2.$ Show that a varible line though $(0,0)$ meets $C$ at one further point, and hence deduce the parametrisation of $C$ given in (2.1). Do the same for $(y^2 = x^3)$ and $(x^3 = y^3-y^4).$

If anyone could show me how to do any of those in (1), I would really appreciate it.. I have no idea how to tackle this problem. Thanks!

2. The first step is to determine what a line through the origin looks like.
The equation of this line will be y = kx for some k. Then:

$y^2 = x^3 + x^2$

Substituting for y:

$(kx)^2 = x^3 + x^2$

$x^3 = (kx)^2 - x^2$

$x^3 = k^2x^2 - x^2$

$x^3 = (k^2 - 1)x^2$

For $x \neq 0$:

$x = k^2 - 1$

And since $y = kx$:

$y = k(k^2 - 1)$