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.