Why does the discriminant of a ODE in the canonical form $a_{11}G^2_x+2a_{12}G_xG_y+\:a_{22}G^2_y=0$ tells us if it is an elliptic/etc equation?

Question

Having

$$a_{11}G^2_x+2a_{12}G_xG_y+\:a_{22}G^2_y=0$$

We divide by $G^2_y$, setting $t=\frac{G_x}{G_y}$ gives us

$$a_{11}t^2+2a_{12}t+\:a_{22}=0$$

The discriminant of this quadratic equation tell us that:

• if it's greater than zero, it is an hyperbolic equation
• if it's equal to zero, it is a parabolic equation
• if it's less than zero, it is an elliptic equation

My question is: Why? Our teacher just gave this piece of information to us, without mathematically proving this

Answer 1

The naming scheme simply refers to the shape of the quadratic curves $$ a_{11}x^2+2a_{12}xy+a_{22}y^2=c $$ Changing coordinates to $\tilde x=x+\frac{a_{12}}{a_{11}}y$ gives $$ \iff a_{11}\tilde x^2-\frac{a_{12}^2-a_{22}a_{11}}{a_{11}}y^2=c $$ which should make the names clear in the elliptic and hyperbolic case. In-between is always the parabolic case, even if this is somewhat degenerate here. It would probably work better if the right side was set to a generic linear expression $b_1x+b_2y+c$.