Discriminant of a Conic Section

$B^2 - 4AC$ is called the discriminant of a conic section. It is an invariant. Depending on the sign of $B^2 - 4AC$, you can tell which of the three conic sections (Ellipse, Hyperbola, Parabola) where $A$, $B$, and $C$ are the coefficients of a rotated Conic Section is described by the equation

$$AX^2 + BXY + CY^2 + DX + EY + F = 0$$

Can anyone explain the logical reasoning to why $B^2 - 4AC$ can identify which Conic Section the equation is?

If $B^2 - 4AC$:

Is greater than zero = Hyperbola

Is less than zero = Ellipse

Is equal to zero = Parabola

There are two ways to prove this:

Formal: You can show, through a bunch of ugly computation, that the expression $B^2-4AC$ is invariant under rotation. So, consider when $B=0$ (in other words, when the conic section's directrix is parallel to one of the axes). It is easy to see that for a hyperbola $-4AC$ is positive, for an ellipse $-4AC$ is negative, and for a parabola $-4AC$ is $0$. For a better worded explanation, go to this link.

Very Informal But Intuitive: Take the equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$. Imagine if $x$ and $y$ were very large numbers. We can forget about $Dx+Ey+F$ because it becomes insignificant when compared to $Ax^2+Bxy+Cy^2=0$. Now, divide by $x^2$:

$$A+B\left(\frac{y}{x}\right)+C\left(\frac{y}{x}\right)^2=0$$

We notice that we now have a quadratic in $\frac{y}{x}$.

This next part is a little hard to explain in words (and my English is sort of bad) but I will try my best.

The number of solutions to this equation represents the number of ways in which the graph of the equation "zooms off towards infinity." Imagine zooming out really far from a graph of a hyperbola. You would only see an "X" formed by two lines (these lines are the asymptotes of the hyperbola). If you solve for $\frac{y}{x}$ in the above equation, you would be solving for the slopes of those lines. Imagine zooming out really far from a graph of a parabola. You would only see one line (the axis of symmetry for the parabola). If you solve for $\frac{y}{x}$ in the above equation, you would be solving for the slope of that line. If you zoomed out really far from a graph of an ellipse, you would see a point.

So, if $A+B\left(\frac{y}{x}\right)+C\left(\frac{y}{x}\right)^2=0$ has two solutions for $\frac{y}{x}$, the equation is a hyperbola. One solution means parabola. Zero solutions means ellipse or circle. The number of solutions corresponds to the sign of $B^2-4AC$.

I sort of like this informal proof because it explains why the discriminant of a conic looks like that of a quadratic.