Exercise 6.5.F in Ravi Vakil's notes: Showing conic $x^2 + y^2=z^2$ in $\mathbb{P}_k^2$ is isomorphic to $\mathbb{P}_k^1$

Question

I have been stuck on Exercise 6.5.F in Ravi Vakil's notes for a little while now, and I would greatly appreciate any hints/comments/solutions!

Let $k$ be a field that is not of characteristic $2$. I want to show that conic $x^2 + y^2=z^2$ in $\mathbb{P}_k^2 = Proj \ k[x,y,z]$ is isomorphic to $\mathbb{P}_k^1 = Proj \ k[u,v]$.

Answer 1

The point $p=(1,0,1)$ is on the conic. Project from $p$ to $\mathbb{P}^1$, a line not passing through $p$, say $x=0$ and show that this gives an isomorphism.

Answer 2

Here's a hint: if $k$ is algebraically closed, you can write down the map explicitly: $$[u:v] \mapsto \left[u^2 - v^2 : 2uv : u^2 + v^2 \right]$$ You can make this rigorous scheme-theoretically by patching together maps on affine open sets.