Mathematics.

conic sections

The General Conic Equation

Analytic Geometry32 minDifficulty6 out of 10

You should know: conic sections, rotation of conics

Overview

Every conic section — circle, ellipse, parabola, or hyperbola, in any position or orientation — is the solution set of a single second-degree equation in two variables, Ax² + Bxy + Cy² + Dx + Ey + F = 0. The cross term Bxy is what allows the conic to be tilted relative to the coordinate axes; when B = 0 the axes of the conic are already aligned with the x- and y-axes. Classifying which type of conic a given equation represents comes down to a single number computed from the coefficients — the discriminant B² − 4AC — while the linear terms Dx + Ey shift the conic's center or vertex away from the origin.

Intuition

Think of the six coefficients as controls on a single flexible curve template. The quadratic coefficients A, B, C control the conic's fundamental shape and tilt — B is the 'shear' term that appears only when the conic's axes are rotated away from horizontal/vertical. The linear coefficients D, E slide the whole curve around the plane without changing its shape, and the constant F adjusts overall scale/position along with them. The discriminant Δ = B²-4AC is invariant under this sliding (translation) and even under rotation of axes, which is exactly why it alone determines conic type regardless of how the curve is centered or tilted.

Formal Definition

Definition

The general second-degree equation in two variables, with A, B, C not all zero, is:

Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
General conic equation
Δ=B2−4AC\Delta = B^2 - 4AC
Discriminant
Δ<0  ⟹  ellipse (circle if A=C, B=0)\Delta < 0 \implies \text{ellipse (circle if } A=C,\ B=0\text{)}
Δ=0  ⟹  parabola\Delta = 0 \implies \text{parabola}
Δ>0  ⟹  hyperbola\Delta > 0 \implies \text{hyperbola}

Notation

NotationMeaning
Δ=B2−4AC\Delta = B^2-4ACDiscriminant of the general conic equation, invariant under rotation and translation of axes
BBCoefficient of the cross term xy; nonzero exactly when the conic's axes are tilted relative to the coordinate axes

Derivation

When B ≠ 0, the conic is tilted; rotating the coordinate axes by an angle θ eliminates the cross term and reveals the type. The rotation formulas are:

x=x′cos⁡θ−y′sin⁡θ,y=x′sin⁡θ+y′cos⁡θx = x'\cos\theta - y'\sin\theta, \qquad y = x'\sin\theta + y'\cos\theta

Coordinate rotation by angle θ

cot⁡(2θ)=A−CB(B≠0)\cot(2\theta) = \frac{A-C}{B} \quad (B \ne 0)

Angle that eliminates the x'y' cross term in the rotated system

A′x′2+C′y′2+D′x′+E′y′+F′=0A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0

After rotation, the equation has no cross term, and the discriminant B^2-4AC is preserved: B'^2 - 4A'C' = B^2-4AC

Properties

Rotation invariance of Δ

Rotating the coordinate axes leaves B2−4AC unchanged, even though A,B,C individually change\text{Rotating the coordinate axes leaves } B^2-4AC \text{ unchanged, even though } A, B, C \text{ individually change}

No cross term means aligned axes

B=0  ⟹  the conic’s axes are parallel to the coordinate axesB = 0 \implies \text{the conic's axes are parallel to the coordinate axes}

Degenerate cases exist

The equation can also describe a point, a line, two lines, or the empty set, depending on the full set of coefficients, not the discriminant alone\text{The equation can also describe a point, a line, two lines, or the empty set, depending on the full set of coefficients, not the discriminant alone}

Completing the square recovers standard form

When B=0, completing the square in x and y separately converts the general form into a translated standard-form conic\text{When } B=0, \text{ completing the square in } x \text{ and } y \text{ separately converts the general form into a translated standard-form conic}

Applications

CAD software classifies quadric curves and surfaces (extended to 3D) by discriminant-like invariants to decide which standard-form fitting routine to apply.

Worked Examples

  1. Identify A=2, B=-4, C=2.

    A=2, B=−4, C=2A=2,\ B=-4,\ C=2

  2. Compute the discriminant.

    Δ=(−4)2−4(2)(2)=16−16=0\Delta = (-4)^2 - 4(2)(2) = 16 - 16 = 0

  3. A zero discriminant indicates a parabola.

    Δ=0  ⟹  parabola\Delta = 0 \implies \text{parabola}

Answer: The equation represents a parabola.

Practice Problems

Difficulty 5/10

Classify the conic 3x² + 3y² - 6xy + 4x - 2 = 0 using the discriminant.

Difficulty 6/10

Complete the square to write x² + y² - 6x + 8y + 21 = 0 in standard form and identify the conic.

Difficulty 7/10

An engineer models a tilted reflector cross-section as x² - 2xy + y² + 4x = 0. Compute the discriminant to determine the reflector's cross-sectional type.

Common Mistakes

Common Mistake

Believing the discriminant alone always tells you whether the conic is degenerate or not.

The discriminant B²-4AC classifies the *type* (ellipse/parabola/hyperbola) assuming the conic is non-degenerate; separately, the full set of coefficients can make the equation describe a point, a pair of lines, or the empty set instead.

Common Mistake

Forgetting to multiply by the leading coefficient when completing the square inside a grouped term, e.g. writing 4(x²-4x+4) but only adding 4 to the other side instead of 4×4=16.

When completing the square inside a factored group like 4(x²-4x+_), the number added inside the parentheses gets multiplied by the outside coefficient before it can be added to the other side of the equation.

Quiz

In the general conic equation, the cross term Bxy is nonzero exactly when:
The discriminant Δ = B² − 4AC of a general conic equation is invariant under:
Which of the following is NOT necessarily true of the general conic equation Ax²+Bxy+Cy²+Dx+Ey+F=0?

Summary

  • Every conic satisfies Ax²+Bxy+Cy²+Dx+Ey+F=0, with the cross term Bxy present exactly when the conic's axes are tilted.
  • The discriminant Δ=B²-4AC classifies non-degenerate conics: negative→ellipse, zero→parabola, positive→hyperbola, and is invariant under rotation of axes.
  • Rotating axes by θ with cot(2θ) = (A-C)/B eliminates the cross term, after which the type is read off directly.
  • Completing the square (when B=0) converts the general equation into a translated standard-form conic, revealing its center/vertex and axis lengths.

References

  1. WebsiteWikipedia — Conic section
  2. WebsiteWikipedia — Matrix representation of conic sections