Let's take $\phi:\mathcal{X}\rightarrow B$ a family of deformations of complex surfaces above the one dimensional disk $B=D_r(0)\subset \mathbb{C}$. Suppose that $X:=\phi^{-1}(0)$ has one ordinary double point $p$, while all other fibers are smooth. So we can take $x,y,z$ complex coordinates around $p$ such that $\phi(x,y,z)=x^2+y^2+z^2$.

Now take the function $m:\widetilde{B}:=D_{\frac{1}{2}}(0)\rightarrow B$, $m(t)=t^2$ and consider the deformation $\widetilde{\mathcal{X}}=\mathcal{X}\times_B \widetilde{B}$. If $\widetilde{p}\in \widetilde{\mathcal{X}}$ is the only point mapping to $p$ then in local coordinates the germ of $\widetilde{\mathcal{X}}$ in $\widetilde{p}$ is isomorphic to the germ in the ordigin of $V(x^2+y^2+z^2-t^2)\subset \mathbb{C}^4$.

I consider the blow up of $\widetilde{\mathcal{X}}$ in $\widetilde{p}$: its exceptional divisor $E$ is a quadric in $\mathbb{P}^3$ and so is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$ by Segre embedding. Finally it shoul come out that $N_{E/\widetilde{\mathcal{X}}}\simeq\mathcal{O}_{\mathbb{P}^1}(-1)\boxtimes\mathcal{O}_{\mathbb{P}^1}(-1)$ and $\mathcal{O}_{\mathbb{P}^1}(-1)\boxtimes\mathcal{O}_{\mathbb{P}^1}(-1)=p_1^*\mathcal{O}_{\mathbb{P}^1}(-1)\otimes p_2^*\mathcal{O}_{\mathbb{P}^1}(-1)$ where $p_1$ and $p_2$ are the two projections from $\mathbb{P}^1\times\mathbb{P}^1$)

My questions are:

1) I have read that for example for the blow up of the origin in $\mathbb{A}^2$, the normal bundle of the exceptional divisor $\mathbb{P}^1$ is $\mathcal{O}_{\mathbb{P}^1}(-1)$, but why is that?

2) In the specific case of my construction in which way can i prove that $N_{E/\widetilde{\mathcal{X}}}$ is actually the box product?