I am trying to learn about infinitesimal deformations and I am particularly looking at Example 1.2.2 (i) from the book Deformations of Algebraic Schemes by Edoardo Sernesi which states the following:
The quadric $Q\subseteq\mathbb{A}^{3}$ of equation $xy-t=0$ defines, via the projection $$\begin{array}{rccc} \pi: & \mathbb{A}^{3} & \longrightarrow & \mathbb{A}^{1} \\ & (x,y,t) & \longmapsto & t. \end{array}$$ a flat family $Q\longrightarrow\mathbb{A}^{1}$. This means that we have the following diagram: $$\require{AMScd} \begin{CD} Q @>{}>> \mathbb{A}^{3}\\ @VVV @VVV{\pi} \\ \mathrm{Spec}(k) @>{}>> \mathbb{A}^{1} \end{CD}$$ where $\pi$ is flat and surjective, and $\mathbb{A}^{1}$ is connected.
Question: How to prove that $\pi$ is flat?
Definition. Let $f:X\longrightarrow S$ be a morphism of schemes. We say $f$ is flat at a point $x\in X$ if the local ring $\mathcal{O}_{X,x}$ is flat over the local ring $\mathcal{O}_{S,f(x)}$. We say $f$ is flat if $f$ is flat at every point of $X$.
Going back to the Example, I have to $\mathfrak{p}\in\mathbb{A}^{3}$, then $\mathcal{O}_{\mathbb{A}^{3},\mathfrak{p}}\cong k[x,y,z]_{\mathfrak{p}}$ and $\mathcal{O}_{\mathbb{A}^{1},\pi(\mathfrak{p})}\cong k[w]_{\left<w-a\right>}$ for some $a\in k$. How prove $k[x,y,z]_{\mathfrak{p}}$ is flat over $k[w]_{\left<w-a\right>}$?
Question: Do you have another idea how to prove that $\pi$ is a flat morphism?
Any hint to solve the doubt will be welcome. Thank you very much.