Identifying conics

Math

Definition

The process of determining whether a second-degree equation represents a circle, ellipse, parabola, or hyperbola. This is done by examining the coefficients of x² and y² terms and the discriminant B² − 4AC of the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0.

Examples

  • x² + y² = 25 is a circle (equal coefficients, same sign)
  • 4x² + 9y² = 36 is an ellipse (different coefficients, same sign)
  • x² − y² = 1 is a hyperbola (opposite signs)
Key Fact

Circle: A = C; Ellipse: A ≠ C, same sign; Hyperbola: A and C opposite signs; Parabola: only one squared term

Related Terms

Ellipses Hyperbolas Parabolas Circles and circumference

Study This Concept

Practice identifying conics with free review games in these units:

Pre-Calculus Unit 8: Conic Sections