Is the discriminant of a second order equation related to the graph of $ax^2+bxy+cy^2+dx+ey+f=0$? Is the discriminant of a second order equation related to the graph of $ax^2+bxy+cy^2+dx+ey+f=0$?

Most people who took precalculus know that $ax^2+bxy+cy^2+dx+ey+f=0$ is the graph of:


*

*An ellipse if $b^2-4ac\lt0$

*A parabola if $b^2-4ac=0$

*A hyperbola if $b^2-4ac\gt0$


Is this related to the fact that the discriminant of $ax^2+bx+c=0$ is $\Delta=b^2-4ac$, which tell us that the equation has 2 solutions if $b^2-4ac\gt0$, one if $b^2-4ac=0$, and none (in the reals) $b^2-4ac\lt0$?

Both solving a quadratic and diagonalising a quadratic form involve completing the square:
$$ 0 = ax^2+bx+c = a\left( x+\frac{b}{2a}\right)^2- \frac{1}{4a}(b^2-4ac) \\
ax^2+bxy+cy^2 = a\left( x+\frac{b}{2a}y\right)^2- \frac{1}{4a}(b^2-4ac)y^2,
 $$
so in both cases the sign of the quantity $b^2-4ac$ determines the nature of solutions.
Geometrically, $ax^2+bxy+cy^2+dx+ey+f=0$ is the intersection of the conic $z=ax^2+bxy+cy^2$ and the plane $ dx+ey+z+f=0 $. Choosing the plane as $z=0$, we can then ask the question: what does the conic
$$ ax^2+bxy+cy^2 = 0 $$
look like? Clearly the point $x=y=0$ is part of it. Does it contain any other points? If $a=0$, it includes the line $y=0$. Now suppose $a \neq 0$. Then there are clearly no points with $y=0$ other than $x=y=0$.
$$ a\frac{x^2}{y^2} + b \frac{x}{y}+c=0 $$
is a quadratic in $x/y$, and has real solutions if and only if $b^2-4ac>0$. Therefore the conclusion is:


*

*If $a=0$, the conic $ ax^2+bxy+cy^2 = 0 $ consists of the lines $y=0$ and $bx+cy=0$.

*If $a \neq 0$, the conic $ ax^2+bxy+cy^2 = 0 $ consists of the lines $x-\alpha_i y=0$, where $\alpha_i$ are the roots of the quadratic $az^2+bz+c=0$.


Therefore the conclusion is that if $b^2-4ac>0$, $ ax^2+bxy+cy^2 = 0 $ consists of 2 lines, if $b^2-4ac=0$, it is one line, and if $b^2-4ac<0$, it is only the point $x=y=0$.
Those are the connections, as far as I can see.
