Conics over fields of characteristic two

I was skimming through my solutions of the exercises in Chapter I of Hartshorne and I found two exercises I haven't been able to fully solve. Both exercises are about conics.

• The first exercise (1.1 c) asks the following: Given an irreducible quadratic polynomial $f$ in $k[x,y]$, show that the affine coordinate ring of $k[x,y]/(f)$ is isomorphic to the coordinate ring of the parabola $y=x^2$ or the hyperbola $xy=1$.

• The second exercise (3.1 c) asks to show that any conic in $\mathbf{P}^2$ is isomorphic to $\mathbf{P}^1$.

Both exercises I've been able to solve whenever the characteristic of the field is different from 2. For the first one I used a brute force calculation where I need to divide by two alot (as morphisms are not really allowed at this point). For the second one I used a symmetric matrix to reduce to the case where the defining polynomial is of the form $F(x,y,z)=ax^2+by^2+cz^2$. This approach also assumes a characteristic different from 2 to construct such a matrix.

My question now is how to do this in the case where the characteristic of the base field is 2. I can't seem to find a way to adapt my current methods to this case. Thanks in advance for any answers!

I've written down details for the following if you need, but I think it's easier to see a sketch then flesh it out on your own. The basic idea is that the "changing coordinates" done below are just $k$-algebra automorphisms of $A(W)$, where $W$ is our conic.

Let $f = a_{11}x^2 + a_{12}xy + a_{22}y^2 + b_1x + b_2y + c$ and $W = Z(f) \subseteq \mathbb{A}^2$. The claim is that $A(W) \cong A(Z(y-x^2))$ if $a_{12} = 0$, and $A(W) \cong A(Z(xy-1))$ if $a_{12} \ne 0$.

If $a_{12} = 0$, then without loss of generality $a_{11} \ne 0$. Changing coordinates $x \mapsto x + \sqrt{a_{22}/a_{11}}y$, we can assume $a_{22} = 0$. $b_2 \ne 0$ since otherwise $f$ is reducible, so change coordinates $y \mapsto y + (b_1/b_2)x$ and $x \mapsto x + \sqrt{c}$ and we are in the case $f = y - x^2$.

If $a_{12} \ne 0$, letting $\alpha$ such that $\alpha^2 + a_{12}\alpha + a_{22} = 0$, change coordinates $x \mapsto x - \alpha y$ and $y \mapsto y + (a_{11}/a_{12})x$ to reduce to the case when $f = a_{12}xy + c$. $c \ne 0$ for otherwise $f$ is reducible, so we are in the case $f = xy - 1$ after rescaling.