Let us consider the conifold singularity $xy-zw=0$ in $\mathbb{C}^4$. By blowing up along the divisor defined by $x=z=0$, we have a small resolution of the conifold with $\mathbb{P}^1$ as the exceptional locus. Here is my question. How can one see that the normal bundle of this $\mathbb{P}^1$ is given by $\mathcal{O}(-1)^{\oplus2}$?

Let us next consider the generalized conifold singularity $xy-(z-w^k)(z+w^k)=0$ in $\mathbb{C}^4$. Again, by blowing up along the divisor defined by $x=z-w^k=0$, we have a small resolution of the generalized conifold with $\mathbb{P}^1$ as the exceptional locus. How can one see that the normal bundle of this $\mathbb{P}^1$ is given by $\mathcal{O}\oplus\mathcal{O}(-2)$?

It would be nice if one can show these normal bundles with explicit coordinates. Thank you in advance.