Preface: C is an
hyperbola if and only if C
has exactly 2 distinct ideal points.
Proof: Suppose $\Delta \gt 0$.
We examine $f(x,y,z)= Ax^2 + Bxy+Cy^2 +Dxz+Eyz+Fz^2 = 0$ when $z =
0$. So ...
$f(x,y,0) = Ax^2 + Bxy+Cy^2 = 0$ (*).
Thus in either case, C
has exactly two distinct ideal points and so C is an hyperbola.