Show that the forms $q(x,y)=x^2-y^2$ and $q'(x,y)=xy$ are isomorphic

Question

This question appears in a textbook, within a section discussing Hyperbolic forms and the Hyperbolic place. Looking at it and at the material over and over, I can't figure out how they're related.

How is this question related to Hyperbolic forms? If it isn't, is there a simple solution not related to that subject?

Answer 1

The level sets of both $q$ and $q'$ are hyperbolas and they are the typical examples of hyperbolic forms. To see that they are isomorphic (over any field in which $2 \neq 0$), consider the change of variables

$$ x = \frac{u + v}{2}, y = \frac{u - v}{2}. $$

Then

$$ q(x,y) = \left( \frac{u + v}{2} \right)^2 - \left( \frac{u - v}{2} \right)^2 = \frac{u^2 + 2uv + v^2 - (u^2 - 2uv + v^2)}{4} = uv = q'(u,v). $$