So I’m trying to prove all conics (i.e. zero sets of irreducible homogeneous polynomials of degree $2$) in $\mathbb{P}^2$ are isomorphic to $\mathbb{P}^1$ (here I work with classical algebraic geometry over a fixed algebraically closed field $k$). My approach is as follows: let $C$ be the conic in question. We may assume $[1:0:0]$ is in $C$. Our polynomial then has the form $$f=\beta x_1^2+\gamma x_2^2+ \delta x_0 x_1 + \epsilon x_1 x_2 +\zeta x_0 x_2.$$ My idea is now to map any point $[x:y]$ in $\mathbb{P}^1$ to the unique intersection of the line through $[1:0:0]$ and $[1:x:y]$ with $C$. Working this out one gets a map $\varphi: \mathbb{P}^1 \rightarrow C$ given by $$[x:y] \mapsto [\beta x^2+ \gamma y^2 + \epsilon xy : -\delta x^2-\zeta x y : -\delta x y - \zeta y^2].$$ Some calculation gives that this is indeed well-defined and bijective (a useful consequence of these calculations is that $[1:0:0]$ is reached by $[-\zeta : \delta]$, which is indeed a well-defined element of $\mathbb{P}^1$ because $f$ is irreducible). It is also clearly a morphism of varieties. I want to understand why $\varphi^{-1}$ is a morphism. I know that on $C - \{[1:0:0] \}$ we have that $\phi^{-1}$ is given by $$[x_0:x_1:x_2] \mapsto [x_1 :x_2],$$ so at least on this open set $\varphi^{-1}$ is a morphism. But I cannot seem to find an open neighborhood of $[1:0:0]$ such that $\varphi^{-1}$ is a morphism there. I thought about maybe proving that the induced map on the level of stalks is an isomorphism, but that seems kind of tricky... Any thoughts? Thanks a lot!
P.S.: I know there is an easier way to do this, one can prove every conic is equivalent to one of the form $x_2^2-x_0x_1$ and then use the $2$-tuple embedding $\mathbb{P}^1 \rightarrow \mathbb{P}^2$, but I'm specifically interested in my approach because the idea is intuitive/geometric.
P.P.S.: Actually I'm having trouble understanding the approach mentioned in "P.S." and I'm interested in understanding it too. The first step in this approach is to see that we can assume $f = x^2+y^2+z^2$ by a change of variables. What change of variables would we need for this?