Conic Sections – Classification and Discriminant Test

Conic Sections – Definition, Classification, and Quick Diagnostic Method

• Overview

  • A conic is defined as the set of curves formed by intersecting a right circular cone with a plane. The four types of curves are: circle, ellipse, parabola, and hyperbola.
  • General second-degree equation (GENERAL FORM) of a conic:
    Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
  • Often, problems are presented as second-degree equations in x and y (quadratic terms present), and we classify the conic from the equation.

• Four Types of Conics (geometric definitions and algebraic criteria)

  • Circle

  • Geometric definition: The plane cuts only one nappe of the cone and is perpendicular to the axis; the image is a circle.

  • Algebraic criteria (general form):

    • Discriminant: B^2 - 4AC < 0
    • For a circle specifically: A = C
      eq 0, ext{ and } B = 0 ext{ with both } x^2 ext{ and } y^2 ext{ terms present}
    • In practice, circle occurs when the quadratic terms are equal in coefficient and the xy-term is absent.

  • Ellipse

  • Geometric definition: The plane cuts only one nappe and is parallel to the base of the cone with an angle (not perpendicular to the axis); image is an ellipse.

  • Algebraic criteria (general form):

    • Discriminant: B^2 - 4AC < 0
    • Same sign for A and C (A C > 0) and both x^2 and y^2 terms present; not a circle when A ≠ C or B ≠ 0.

  • Parabola

  • Geometric definition: The plane is parallel to a generator and perpendicular to the base; the curve is a parabola.

  • Algebraic criteria (general form):

    • Discriminant: B^2 - 4AC = 0
    • Only one quadratic term is present (either Ax^2 or Cy^2); i.e., either A = 0 or C = 0 (in the unrotated case). If B ≠ 0, the parabola can be rotated, but the discriminant condition still holds: B^2 - 4AC = 0.

  • Hyperbola

  • Geometric definition: The plane cuts both nappes and is parallel to two generators; the intersection is a hyperbola.

  • Algebraic criteria (general form):

    • Discriminant: B^2 - 4AC > 0
    • A and C have opposite signs (AC < 0) and both x^2 and y^2 terms are present.

• How to identify the type of a conic from a given quadratic equation

  • Primary method: compute the discriminant
  • ext{Discriminant} = B^2 - 4AC
  • If discriminant < 0 → ellipse or circle (depending on A, C, B)
  • If discriminant = 0 → parabola
  • If discriminant > 0 → hyperbola
  • Secondary method: inspect the quadratic terms
  • Check which quadratic terms appear: x^2, y^2, xy
  • Use signs and equalities of coefficients to distinguish circle/ellipse from parabola/hyperbola
  • Important note: Always transform the equation to the GENERAL EQUATION form before applying the discriminant test.

• Worked examples (classroom-style discriminant checks)

  • Example 1 (Circle)

  • Given: x^2 + y^2 - 3x + 4 = 0

  • Parameters: A = 1,
    d B = 0,
    C = 1

  • Discriminant: B^2 - 4AC = 0^2 - 4(1)(1) = -4 < 0

  • Quadratic terms: x^2, y^2 with equal coefficients and no xy-term → circle (A = C, B = 0).

  • Example 2 (Ellipse)

  • Given: 3x^2 - 9x = -2y^2 - 10y + 6

  • Rearrange to general form: 3x^2 - 9x + 2y^2 + 10y - 6 = 0

  • Parameters: A = 3,
    B = 0,
    C = 2

  • Discriminant: B^2 - 4AC = 0 - 4(3)(2) = -24 < 0

  • Quadratic terms have same sign (A and C both positive) and both x^2 and y^2 appear → ellipse.

  • Example 3 (Parabola)

  • Given: x^2 - 3x - y + 7 = 0

  • Parameters: A = 1,
    B = 0,
    C = 0

  • Discriminant: B^2 - 4AC = 0 - 0 = 0

  • Only one quadratic term present (x^2) → parabola.

  • Example 4 (Hyperbola)

  • Given: 2x^2 - 3y^2 + 2x - y + 22 = 0

  • Parameters: A = 2,
    B = 0,
    C = -3

  • Discriminant: B^2 - 4AC = 0 - 4(2)(-3) = 24 > 0

  • Quadratic terms have opposite signs (A and C opposite) → hyperbola.

• Quick reference checklist (from the lesson materials)

  • Always transform to the general form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 before classification.
  • Compute oxed{B^2 - 4AC} to decide the type:
  • < 0: circle or ellipse (same sign for A and C; B often 0 for standard axes-aligned cases)
  • = 0: parabola (only one of the quadratic terms present in the unrotated case; can be rotated if B ≠ 0 but still discriminant 0)
  • > 0: hyperbola (A and C have opposite signs; both x^2 and y^2 present)
  • Interpret quadratic terms: presence of x^2 and y^2 terms, their coefficients, and the xy-term influence whether the conic is a circle, ellipse, parabola, or hyperbola.

• Context and learning outcomes (as stated in the course materials)

  • Content Standard: Understand key concepts of conic sections and systems of nonlinear equations.
  • Performance Standard: Model situations using conic sections and nonlinear equations; solve problems accurately.
  • Learning Competencies:
  • Illustrate different types of conic sections: circle, ellipse, parabola, hyperbola.
  • Recognize the equation and important characteristics of each conic type.

• Summary note

  • The four conics are determined by the signs and presence of the quadratic terms in the general second-degree equation, with the discriminant B^2 - 4AC serving as the primary diagnostic tool.
  • Transform any given equation to its general form first, then apply the discriminant and term inspection to classify the curve.