Using the general equation of the conics we can analyze what type of conical section we can get:
[tex]Ax^2+Bx\cdot y+Cy^2+Dx+Ey+F=0[/tex]
By changing the values of any of the constants, the shape of the corresponding conic will also change. For the case of the hyperbola
[tex]\begin{gathered} B^2-4\cdot A\cdot C>0 \\ \end{gathered}[/tex]
For this inequality to be fulfilled, we can guarantee that the second term is positive, since the second term is negative (-4AC), this must be less than zero, and by law of signs it becomes positive and the inequality is fulfilled.
[tex]A\cdot C<0_{}\to\text{True}[/tex]
