Identify and analyze conic sections from the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. Find the type, center, foci, vertices, eccentricity, and standard form for circles, ellipses, parabo...
The **Conic Sections Calculator** takes the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 and identifies which conic section it represents — circle, ellipse, parabola, or hyperbola — then extracts all key geometric properties. Enter the six coefficients and the tool instantly calculates the discriminant, determines the curve type, and completes the square to find the center, foci, vertices, semi-axes (or radius), eccentricity, and directrix.
Conic sections appear throughout mathematics and science. Planetary orbits are ellipses (Kepler's first law), satellite dish reflectors use parabolic curves, cooling tower profiles follow hyperbolas, and circles are everywhere in engineering. Understanding how to classify and parameterize these curves from their equation is a core skill in analytic geometry, precalculus, and multivariable calculus.
The discriminant B² − 4AC is the key classifier: negative for ellipses and circles, zero for parabolas, and positive for hyperbolas. The calculator visualises this with a color-coded type badge and eccentricity bar — eccentricity of 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, and greater than 1 for a hyperbola. A comparison table summarises the conditions, standard forms, and properties side by side.
Eight presets cover the most common textbook examples: the unit circle, standard ellipses and hyperbolas, axis-aligned parabolas, and conics with translated centres. Click any preset, review the properties, then modify the coefficients to explore how the curve changes.
Conic Sections Calculator — Circle, Ellipse, Parabola, Hyperbola helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Conic Type, Center, Foci in one pass.
Discriminant: Δ = B² − 4AC. Circle: Δ < 0 and A = C, B = 0. Ellipse: Δ < 0 and A ≠ C. Parabola: Δ = 0. Hyperbola: Δ > 0. Eccentricity: e = c/a where c² = a² − b² (ellipse) or c² = a² + b² (hyperbola). For a circle, e = 0. For a parabola, e = 1.
Result: Computed from the entered values
Using A=1, B=0, C=1, D=0, the calculator returns Computed from the entered values. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This calculator is tailored to conic sections calculator — circle, ellipse, parabola, hyperbola workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
A conic section is the curve formed by intersecting a plane with a double-napped cone. Depending on the angle, the result is a circle, ellipse, parabola, or hyperbola.
The discriminant Δ = B² − 4AC determines the type: if Δ < 0 the conic is an ellipse (or circle when A = C); if Δ = 0 it is a parabola; if Δ > 0 it is a hyperbola. Use this as a practical reminder before finalizing the result.
Eccentricity (e) measures how much a conic deviates from a circle. A circle has e = 0, an ellipse 0 < e < 1, a parabola e = 1, and a hyperbola e > 1.
A non-zero B coefficient means the conic is rotated relative to the coordinate axes. You can eliminate the Bxy term by rotating the axes through the angle θ = ½ arctan(B/(A − C)).
When the equation factors into two lines (or represents a single point or the empty set), it's called degenerate. Examples: x² − y² = 0 → two intersecting lines, x² + y² = 0 → just the origin.
First identify a (semi-major) and b (semi-minor) from the standard form. Then c = √(a² − b²). The foci lie along the major axis at distance c from the center.