= Solution
The two pairs of regular functions
$$
[x:w]
\quad\hbox{and}\quad
[z:y]
$$
define maps to the <projective line> on the loci where their respective coordinates do not vanish simultaneously. Those loci cover $Y$, since simultaneous failure would force $x=y=z=w=0$. On their overlap the equation $xy=zw$ says that the two projective points are equal. They therefore glue to a morphism
$$
f:Y\longrightarrow\mathbb P_k^1.
$$
Let $[A:B]$ be the homogeneous coordinates on $\mathbb P_k^1$. The pullback of the <hyperplane divisor> $V(A)$ has local equation $x$ on the first chart and $z$ on the second. It is therefore exactly $D_Y=V(x,z)$. Compatibility of the <pullback of a sheaf of modules> with the <line bundle associated to a divisor> gives
$$
f^*\mathcal O_{\mathbb P_k^1}(1)
\cong\mathcal O_Y(D_Y).
$$
This is the <ruling morphism of the punctured three-dimensional affine quadric cone> associated with $D_Y$.
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