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:
Substituting for y:
For :
And since :
(2.1) Examples of parametrised cubics. Some plane cubic curves can be parametrised, just as the conics:
Nodal cubic is the image of the map given by
Cuspidal cubic. is the image of given by
(1) Let Show that a varible line though meets at one further point, and hence deduce the parametrisation of given in (2.1). Do the same for and
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!