I am trying to parametrize the following curves:
$F(z,x,y) = (x-y)^7 - z^5x^2, \quad G(z,x,y) = (x-y)^3 - zx^2$.
I first try to find a parametrization for $g = G(1,x,y)$. I am trying to do it along the lines through the origin. Hence $y = xt$. For $g$ I get the following:
$(x-xt)^3 - x^2 = x^3(1-t)^3 - x^2 \implies x^3(1-t) = x^2 \iff x = \frac{1}{(1-t)^3} \implies y = \frac{t}{(1-t)^3}$,
which will give me the parametrization $[t_0,t_1] \mapsto [(t_1-t_0)^3 : t_0^3 :t_0^2t_1]$.
However, doing the same with $f = F(1,x,y)$ I will get $x^7(1-t) = x^2 \implies x = \frac{1}{(1-t)^{7/5}} \implies y = \frac{t}{(1-t)^{7/5}} $. But here I don't know how to get the parametrization from that. How can I proceed from there?