In my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ($k$ is algebraically closed of course, and let's say characteristic 0 for simplicity), say $X \subseteq \mathbb{P}^2_A$, flat over $A$, is defined by a single degree 2 polynomial. More precisely, the ideal in $A[x,y,z]$ corresponding to $X$ is principal, of degree 2. 

There's a somewhat complicated story behind this question; in particular this exercise [has been asked](https://mathoverflow.net/questions/292546/exercise-1-1-c-in-hartshornes-deformation-theory) on overflow, and the conclusion was that the question is wrong, and is only true **locally**. To me, that seems like it should be true for any finitely generated local $k$-algebra. However, I came up with the following example over $A = k[t]_{\langle t-1 \rangle}$; consider $$ \operatorname{Proj}A[x,y,z]/\langle tyz - x^2, yz - tx^2 \rangle.$$ Obviously the only closed fibre is $\operatorname{Proj} k[x,y,z]/yz = x^2$, but I am in the awkward situation of not being able to show that the ideal of the $X$ is principally generated, yet also not being able to show it is not flat. Indeed, one can show that $$(t+1)(yz-x^2) = 0$$ and since $t+1$ is invertible in $A$, it follows $yz-x^2 = 0$. However, I am not able to show that $tyz - x^2$ is in the ideal $\langle yz - x^2 \rangle$, and not able to find any torsion element either: obviously we have $$(t-1)(yz - x^2) = 0,$$ but as mentioned $yz-x^2$ is already zero, so this doesn't help!

And on that note, could I please have some hints on how to approach (the local version of) the exercise? Unfortunately the quoted overflow page only has the answer as to why it's **not** true.

Thanks in advance!