Plane

Given equation:

Rewriting, we get:

Now simplify:

Now, observe that for , we must have:

Now substitute back to find :

(since )

But, the equation has infinitely many solutions:

Each such equation defines a branch of the hyperbola

Each branch is asymptotic to lines of slope ±1.

Hence, the correct answer is: (C) infinitely many lines with slope ±1.