Question

In the standard equation for a conic section Ax + Bxy + y2 + Dx +Ey + F = 0, if AC < 0, the conic section in question is a hyperbola.TrueFalse

Answer 1

Using the general equation of the conics we can analyze what type of conical section we can get:

`Ax^2+Bx\cdot y+Cy^2+Dx+Ey+F=0`

By changing the values of any of the constants, the shape of the corresponding conic will also change. For the case of the hyperbola

`\begin{gathered} B^2-4\cdot A\cdot C>0 \\ \end{gathered}`

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.

`A\cdot C<0_{}\to\text{True}`