Situation:
Consider the curves $ C_1 = x^2 + y^2 +x^2y^2 = 1 $ and $C_2 = w^2 = 1 - z^4 .$
Define a map $\lambda$ from $C_1$ to $C_2$ with $\lambda(x,y) = (x,(1+x^2)y)$.
This map should be bijective. injective was ok, but I failed to show that this map is surjective.
Any point $(z,w)$ in $C_2$ is the image of the point $(x,y)=(z,\frac w {1+z^{2}})$. All you have to do is to verify the condition $x^{2}+x^{2}+x^{2}y^{2}=1$ using the condition $w^{2}=1-z^{4}$.