Conic Sections Calculator

Identify and analyze conic sections from the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0.

Enter Coefficients

General form: Ax² + Bxy + Cy² + Dx + Ey + F = 0

x² + y² - 6x - 4y + 4 = 0

Result

Conic Type
Circle
Discriminant (B²-4AC)--
Center--
Standard Form--
Key Parameter--
Eccentricity--

Step-by-Step Solution

Ax² + Bxy + Cy² + Dx + Ey + F = 0

Understanding Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane at various angles. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas. They were first studied by the ancient Greeks, particularly Apollonius of Perga around 200 BCE, and remain fundamental objects in mathematics, physics, and engineering.

Every conic section can be described by the general second-degree equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0. The type of conic is determined by the discriminant B² - 4AC.

Classification Using the Discriminant

The discriminant Δ = B² - 4AC determines the type of conic section:

  • Δ < 0 and A = C, B = 0: Circle
  • Δ < 0 and A ≠ C (or B ≠ 0): Ellipse
  • Δ = 0: Parabola
  • Δ > 0: Hyperbola

Standard Forms of Conic Sections

Circle

All points equidistant from a center point. Eccentricity = 0.

(x-h)² + (y-k)² = r²

Ellipse

Sum of distances to two foci is constant. Eccentricity between 0 and 1.

(x-h)²/a² + (y-k)²/b² = 1

Parabola

Equal distance from focus and directrix. Eccentricity = 1.

(y-k)² = 4p(x-h)

Hyperbola

Difference of distances to two foci is constant. Eccentricity > 1.

(x-h)²/a² - (y-k)²/b² = 1

General Equation

The most general form for all conics.

Ax² + Bxy + Cy² + Dx + Ey + F = 0

Discriminant Test

Classify the conic using the discriminant of the general equation.

Δ = B² - 4AC

Completing the Square

To convert from general form to standard form, the technique of completing the square is used. This involves grouping x and y terms separately, factoring out the leading coefficients, and adding/subtracting constants to form perfect square trinomials.

Example: Identifying a Circle

Given x² + y² - 6x - 4y + 4 = 0:

  1. Group terms: (x² - 6x) + (y² - 4y) = -4
  2. Complete the square: (x² - 6x + 9) + (y² - 4y + 4) = -4 + 9 + 4
  3. Factor: (x - 3)² + (y - 2)² = 9
  4. This is a circle with center (3, 2) and radius 3.

Eccentricity

The eccentricity (e) measures how much a conic deviates from being circular:

  • Circle: e = 0 (perfectly round)
  • Ellipse: 0 < e < 1 (oval shape; closer to 0 means more circular)
  • Parabola: e = 1 (open curve)
  • Hyperbola: e > 1 (two separate branches)

Applications of Conic Sections

  • Planetary orbits: Planets move in elliptical orbits (Kepler's First Law).
  • Satellite dishes: Parabolic reflectors focus signals to a single point.
  • Optics: Elliptical mirrors focus light from one focus to the other.
  • Architecture: Arches, domes, and bridges use conic curves for strength.
  • Navigation: Hyperbolic navigation systems (LORAN) use hyperbolic curves.
  • Projectile motion: Objects thrown near Earth follow parabolic trajectories.