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 ...
$f(x,y,0) = Ax^2 + Bxy+Cy^2 = 0$ (*).
Case 1: $A = 0.$
Case 2: $A \ne 0.$
Thus in either case,
C
has exactly two distinct ideal points and so
C
is an hyperbola.