Relation between hyperbolic plane and its definition?

Question

I am reading a book that states a hyperbolic plane is a quadratic module with a basis of two vectors $x, y$ that are isotropic and such that the associated symmetric bilinear form on $x, y \neq 0.$ That is, $x.y \neq 0.$ However, why is this called a hyperbolic plane?

This may be a very general question but how do the conditions specified in the definition affect the 'visualization' of the vector space? Is it related at all to the geometry of a hyperbola? Any answers are appreciated, thank you.

Relevant information: A quadratic module is a pair $(V, Q)$ such that $V$ is a finite dimensional vector space over a field of characteristic $\neq 2$ and $Q: V \rightarrow F$ is such that $Q(ax) = a^2Q(x)$ and the map $Q(x + y) - Q(x) - Q(y)$ is bilinear. The map is denoted as $x.y = Q(x + y) - Q(x) - Q(y).$ Furthermore, an isotropic vector is a vector such that $x.x = 0.$

Answer 1

Bit long for a comment. In 2005, T.Y. Lam published Introduction to Quadratic Forms over Fields. On page 9 is Theorem 3.2. Then page 10 begins with

The isometry class of a 2-dimensional quadratic space satisfying the conditions in Theorem 3.2 is called the "hyperbolic plane" (presumably because the graphs of the equations $X_1 X_2 = d$ are called hyperbolas).