General second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0 — discriminant classification: Foundations

This page is the "unpack your toolbox before the mission" page for the parent topic. Everything the parent note assumed you already knew, we build here, one brick at a time.

0. What does an equation draw?

Before any symbols, one idea: an equation in $x$ and $y$ is a rule that says yes or no to points.

Figure — General second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0 — discriminant classification

We need this because the whole topic talks about "the curve of the equation" — that phrase only makes sense once you see an equation as a yes/no filter that paints a set of points.

1. Powers: $x^{2}$ , and what "second-degree" means

The six pieces of the master equation, sorted by degree:

Term	Degree	Job
$A x^{2}, B x y, C y^{2}$	2	the quadratic part — sets the shape
$D x, E y$	1	the linear part — slides the shape around
$F$	0	the constant — sizes/positions it

2. Coefficients: what $A, B, C, D, E, F$ are

The rule $(A, B, C) \neq = (0, 0, 0)$ just means: at least one degree-2 dial is non-zero — otherwise there are no squared terms and the picture is only a line, not a conic.

3. The cross term $B x y$ and the idea of "tilt"

This is the piece most people have never met, so it gets its own picture.

We need this because the parent's whole "rotation invariant" story is about removing the $x y$ term — you cannot follow that story without first knowing $B x y$ = tilt.

4. The two ways a quadratic form can "point": definite vs indefinite

Strip the equation down to just its shape-setting core, $A x^{2} + B x y + C y^{2}$ . This expression is a quadratic form — a recipe that eats a point $(x, y)$ and spits out a single number.

Before the picture, one more piece of vocabulary, because the figure shows curves of the form $Q = k$ .

Intuition Three personalities of

$Q$ — and the three conics

Always same sign (positive everywhere except the origin, or negative everywhere): the level curve $Q = k$ is a closed loop → the ellipse family. We call $Q$ definite.
Flips sign (positive in some directions, negative in others): the level curve breaks into two open branches → the hyperbola family. We call $Q$ indefinite.
The knife-edge in between ( $Q$ never goes negative but hits exactly $0$ along one whole direction): $Q$ is semi-definite. Here a subtlety matters — the level curves of the pure form $Q = k$ are just parallel lines (or a doubled line), not yet a parabola. You only get a genuine parabola once the linear terms $D x + E y$ are switched back on; the $D x + E y$ part bends those parallel lines into the U-shape. Without it, the semi-definite case collapses to line-pairs — one of the degenerate outcomes.

A note on the word "degenerate": here we describe $Q$ in the borderline $B^{2} - 4 A C = 0$ case as semi-definite. Reserve the word degenerate for the collapsed curves (point, single line, line-pair, empty) discussed in Degenerate Conics — the two words are not synonyms, even though the semi-definite form is where degenerate line-pairs can appear.

See Quadratic Forms and Eigenvalues for the deeper machinery, and Discriminant of a Quadratic for where the word "discriminant" first comes from.

5. Signs and the " $= 0$ " on the right

Everything hinges on signs ( $+$ and $-$ ) and on comparing to zero, so make these explicit.

6. Prerequisite map

Each foundation box feeds the final topic node. Notice the two rivers that must both arrive: what the equation draws (left) and which sign the discriminant gives (right).

Once these foundations are solid, the standard shapes live in Ellipse Standard Equation, Parabola Standard Equation, Hyperbola Standard Equation, and Circle as Special Ellipse. When a conic collapses, see Degenerate Conics. The tilt-removal procedure is Rotation of Axes.

Equipment checklist

Test yourself — reveal only after answering out loud.

What does the notation

(x, y)

describe, and where do you measure it from?

A point;

x

= signed distance right/left,

y

= signed distance up/down, both measured from the origin

(0, 0)

along the axes.

What does it mean for a point to be "on the curve" of an equation?

Plugging its

x, y

into the equation makes both sides equal (it passes the yes/no filter).

What is a conic, in one sentence?

The curve got by slicing a cone with a plane — circle, ellipse, parabola, or hyperbola — equivalently the non-collapsed curve of a second-degree equation.

What is the degree of the term

x y

, and why?

Degree

2

— it has two variable-factors, one

x

and one

y

Which variable-term is

B

the coefficient of?

The cross term

x y

(NOT

x

— that one is

D

What does the presence of a

B x y

term tell you about the conic's orientation?

It is tilted (rotated) relative to the

x

–

y

axes.

What is a quadratic form, in one sentence?

The shape-setting part

A x^{2} + B x y + C y^{2}

: a rule that takes a point and returns one number.

What is a level curve

Q = k

, and how does it relate to the full equation?

The set of points where the form outputs the fixed value

k

; the full conic is

Q = - (D x + E y + F)

, i.e. a slanted/shifted slice of the same landscape.

What is the difference between a definite and an indefinite quadratic form, picture-wise?

Definite = output keeps one sign → closed-loop level curves (ellipse); indefinite = output flips sign → two-branch level curves (hyperbola).

In the semi-definite (

B^{2} - 4 A C = 0

) case, what does the pure form give, and what is needed for a parabola?

The pure form's level curves are parallel lines (or a doubled line); a genuine parabola needs the linear terms

D x + E y

switched on to bend them.

Write the discriminant formula and its three sign-verdicts.

B^{2} - 4 A C

; negative → ellipse, zero → parabola, positive → hyperbola.

Why is "semi-definite" not the same word as "degenerate"?

Semi-definite describes the borderline form (

B^{2} - 4 A C = 0

); degenerate describes a collapsed curve (point, line, line-pair, empty).

Why is the master equation written with "

= 0

" on the right?

Any equation can be rearranged so everything sits on one side against zero; it is just the standard tidy form and loses no curves.

What must be true of

(A, B, C)

for the equation to be a genuine conic and not just a line?

They cannot all be zero — at least one degree-2 term must survive.