Why $b^2-4ac$ as determinant?

Question

I am curious why $b^2-4ac$ is used as a determinant of a conic section?

Like why this specific expression is chosen, why the value is always greater, lesser or equal to zero for hyperbola, ellipse and parabola and how does the value prove the type of conic section.

A detailed answer would be of real help.

Answer 1

If you have a conic $$ax^2+bxy+cy^2=d$$ then you can complete the square,

$$a\left(x+\frac{b}{2a}y\right)^2-\frac{b^2-4ac}{4a}y^2=d$$ or

$$4a^2\left(x+\frac{b}{2a}y\right)^2-Dy^2=4ad$$

Since $a^2$ is positive we see that the form of the conic is determined by the sign of $D$.