terican

Last verified · v1.0

Calculator · math

Conic Sections Calculator

Enter coefficients A through F to classify any conic section. Computes discriminant Δ=B²−4AC, conic type, and center coordinates instantly.

FreeInstantNo signupOpen source

Inputs

Conic Result

—

Explain my result

0/3 free

Get a plain-English breakdown of your result with practical next steps.

Conic Result—

The formula

How the
result is
computed.

Understanding the General Conic Equation

A conic section arises from the intersection of a plane with a double-napped right circular cone. The general second-degree equation in two variables captures every conic — circle, ellipse, parabola, and hyperbola — in a single unified algebraic form:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

Six real coefficients (A, B, C, D, E, and F) completely determine the shape, orientation, size, and position of the resulting curve. This standardized representation, as documented in Richland Community College's authoritative Conic Sections reference, forms the algebraic foundation for all second-degree curve analysis in analytic geometry.

The Discriminant: Δ = B² − 4AC

The most powerful diagnostic metric for classifying any conic section is the discriminant:

Δ = B² − 4AC

Comparing Δ to zero immediately identifies the conic type without requiring any coordinate transformation or algebraic reduction to standard form. This classification rule derives from the theory of invariants under rotation of axes, as detailed by Rice University's RUSMP guide, Exploring Conics with Graphing Technology:

  • Δ < 0 — Ellipse (reduces to a circle when B = 0 and A = C)
  • Δ = 0 — Parabola
  • Δ > 0 — Hyperbola

Degenerate cases — a single point, two intersecting lines, or the empty set — also satisfy one of these discriminant conditions but fail the separate non-degeneracy test on the full 3×3 coefficient matrix determinant.

Variable Definitions

  • A — Coefficient of x². Controls horizontal curvature; must be nonzero for an x-directed conic.
  • B — Coefficient of the xy cross-term. Any nonzero B indicates the conic is rotated with respect to the coordinate axes.
  • C — Coefficient of y². Controls vertical curvature; must be nonzero for a y-directed conic.
  • D — Coefficient of the linear x term. Shifts the center horizontally from the origin.
  • E — Coefficient of the linear y term. Shifts the center vertically from the origin.
  • F — Constant term. Scales the overall figure and governs whether a real curve exists.

Center Coordinates for Non-Rotated Conics

When B = 0, the axes of symmetry align with the coordinate axes. The center (h, k) satisfies two linear equations obtained by setting the partial derivatives of the quadratic terms equal to zero:

  • h = −D / (2A) (valid when A ≠ 0)
  • k = −E / (2C) (valid when C ≠ 0)

For a parabola where only one second-degree coefficient is nonzero, only one center coordinate is well-defined; the other describes the axis of symmetry's linear offset.

Worked Examples

Example 1 — Circle

Equation: x² + y² − 4x + 6y − 3 = 0. Coefficients: A = 1, B = 0, C = 1, D = −4, E = 6, F = −3.

Discriminant: Δ = 0² − 4(1)(1) = −4 < 0 → Ellipse. Since B = 0 and A = C = 1, this degenerates to a circle. Center: h = 4/2 = 2, k = −6/2 = −3. Center at (2, −3); radius = √16 = 4.

Example 2 — Hyperbola

Equation: x² − y² − 1 = 0. Coefficients: A = 1, B = 0, C = −1, D = 0, E = 0, F = −1.

Discriminant: Δ = 0 − 4(1)(−1) = 4 > 0 → Hyperbola. Vertices at (±1, 0); asymptotes y = ±x.

Example 3 — Rotated Ellipse

Equation: x² + xy + y² − 1 = 0. Coefficients: A = 1, B = 1, C = 1.

Discriminant: Δ = 1 − 4(1)(1) = −3 < 0 → Ellipse. The nonzero B = 1 confirms a rotation of 45° relative to the coordinate axes.

Real-World Applications

  • Orbital mechanics — Planetary orbits are ellipses with the Sun at one focus (Kepler's First Law; eccentricity 0 < e < 1).
  • Satellite dish design — Parabolic reflectors focus all incoming parallel signals to a single focal point.
  • Cooling towers — Hyperboloid structures derive structural efficiency from their hyperbolic cross-sections.
  • Optics and lenses — Hyperbolic and elliptic lens profiles eliminate spherical aberration in high-precision instruments.
  • Projectile motion — Under uniform gravity and no air resistance, a projectile's path traces a true parabola.

Reference

Frequently asked questions

How does the conic sections calculator determine the type of conic?
The calculator computes the discriminant Δ = B² − 4AC from the six entered coefficients. If Δ < 0, the conic is an ellipse or circle. If Δ = 0 exactly, it is a parabola. If Δ > 0, it is a hyperbola. This three-way test requires no coordinate transformation and classifies correctly even for rotated conics where B ≠ 0.
What does a nonzero B coefficient mean in the general conic equation?
A nonzero B coefficient on the xy cross-term signals that the conic is rotated relative to the standard coordinate axes. Removing the rotation requires substituting x and y with rotated variables at angle θ = (1/2) arctan(B / (A − C)). Until that substitution is applied, center coordinates computed via h = −D/(2A) and k = −E/(2C) are not geometrically meaningful.
How do you find the center of an ellipse or hyperbola from the general equation?
For any non-rotated conic where B = 0, the center coordinates are h = −D / (2A) and k = −E / (2C). As a concrete example, the ellipse 4x² + 9y² − 16x + 18y − 11 = 0 has A = 4, C = 9, D = −16, E = 18. This gives h = 16/8 = 2 and k = −18/18 = −1, placing the center at (2, −1).
What is the difference between an ellipse and a circle in the general conic form?
Both an ellipse and a circle produce a negative discriminant Δ < 0. A circle is the special subcase where B = 0 and A = C simultaneously. For instance, x² + y² − 6x + 4y + 4 = 0 has A = C = 1 and B = 0, confirming a circle. When A ≠ C but Δ remains negative, the figure is a true ellipse with two unequal semi-axes a and b.
Can the conic sections calculator handle rotated conics where B is nonzero?
Yes, for classification purposes. The discriminant Δ = B² − 4AC is invariant under any rotation of axes, so it reliably identifies the conic type regardless of rotation. However, the center formulas h = −D/(2A) and k = −E/(2C) assume the axes of symmetry are parallel to the coordinate axes. For rotated conics (B ≠ 0), those formulas yield the center only after performing a rotation of axes to eliminate the cross-term.
What happens when all second-degree coefficients are zero in the conic equation?
If A = B = C = 0 simultaneously, the equation Dx + Ey + F = 0 describes a straight line, which is not a conic section. The discriminant formula returns Δ = 0, but this result is meaningless in that context. Degenerate conics — a single point, two intersecting lines, or the empty set — also appear as edge cases when the determinant of the associated 3×3 symmetric matrix equals zero while A, B, or C remains nonzero.