One parameter, four shapes
Slice a cone at different angles and you get every conic. The eccentricity singlehandedly labels them: 0 (circle), between 0 and 1 (ellipse), 1 (parabola), above 1 (hyperbola).
Eccentricity code
Conic Equation Finder — Circle / Parabola / Ellipse / Hyperbola
Inputs
Results
Standard form
General quadratic form
Ax² + Bxy + Cy² + Dx + Ey + F = 0
Key features
Preview
Tip: You can pan with drag and zoom with your trackpad or browser zoom. Preview is illustrative (numerical sampling)—extreme or huge-scale inputs may be auto-clipped.
What this tool computes
- Circle — from 3 points, center & radius, or diameter endpoints.
- Parabola — from focus & directrix (any orientation), vertex & point, or 3 points (vertical or horizontal).
- Ellipse — from two foci & sum of distances (2a), or center with semi-axes and rotation.
- Hyperbola — from two foci & difference (2a), or center with semi-axes and rotation.
Outputs both a human-friendly standard form and the general quadratic form Ax² + Bxy + Cy² + Dx + Ey + F = 0, plus parameters like center/vertex, radius/axes, focus/foci, directrix, eccentricity, and rotation.
5 Fun Facts about Conics
Reflective superpower
Ellipses whisper secrets: sound from one focus bounces to the other. Hyperbolas do the opposite—rays from one focus reflect as if they started at the other.
Optics magic
Parabola = equal distance club
Every point on a parabola sits exactly as far from the focus as from the directrix. That makes satellite dishes beam signals back to one hot spot.
Dish physics
General form is a matrix
The six numbers in Ax² + Bxy + Cy² + Dx + Ey + F = 0 are just a 3×3 symmetric matrix in disguise. Rotate the axes and you’re basically diagonalizing that matrix.
Linear algebra lens
Orbit DNA
Kepler’s first law: planets trace ellipses with the Sun at one focus. Comets? Often hyperbolas or very stretched ellipses—conics run the solar system.
Space geometry
How to Find Equations of Circles, Parabolas, Ellipses, and Hyperbolas
This tool computes the equation of a circle, parabola, ellipse, or hyperbola from friendly inputs such as points, foci, a directrix, or center–radius parameters. For every case it returns a human-readable standard form as well as the general quadratic form
Ax² + Bxy + Cy² + Dx + Ey + F = 0. It also reports useful geometric features—center or vertex, axes lengths, rotation angle, eccentricity, and (for parabolas) focus and directrix.
Circle
A circle with center (h,k) and radius r has equation
(x − h)² + (y − k)² = r². You can provide the center and radius directly, two endpoints of a diameter, or any three non-collinear points. When you enter three points, the tool solves the perpendicular bisector system to recover (h,k) and r. The general form appears with A = C and B = 0.
Parabola
A parabola is the set of points equidistant from a focus and a directrix. In a rotated local frame (x′, y′) aligned to the axis, the standard form is x′² = 4p·y′, where p is the focal length. You can define a parabola by focus and directrix (ax + by + c = 0), by vertex plus a sample point and axis angle, or by fitting three points to a vertical (y = ax² + bx + c) or horizontal (x = ay² + by + c) model. The tool derives a rotation angle automatically when needed and converts back to the global coordinates to produce Ax² + Bxy + Cy² + Dx + Ey + F = 0.
Ellipse
An ellipse has two foci F₁ and F₂; the sum of distances to the foci is constant and equal to 2a. With center (h,k), semi-major axis a, semi-minor axis b, and rotation angle θ, the axis-aligned form is x′²/a² + y′²/b² = 1. If you provide foci and the sum, the tool infers a, b, and θ (since c = ‖F₁F₂‖/2 and b = √(a² − c²)). Eccentricity is reported as e = c/a, with 0 < e < 1.
Hyperbola
A hyperbola satisfies a constant difference of distances to two foci, equal to 2a. In its principal frame the standard forms are x′²/a² − y′²/b² = 1 or y′²/a² − x′²/b² = 1. Given foci and the required difference, the tool computes a, b = √(c² − a²), center, and rotation angle from the foci line. Eccentricity is e = c/a > 1.
About the General Quadratic Form
Any conic in the plane can be written as Ax² + Bxy + Cy² + Dx + Ey + F = 0. The mixed term Bxy indicates rotation. This tool constructs the exact standard form in a local axis frame and then applies a rotation–translation transform to recover the global coefficients. For readability, coefficients are normalized and tiny rounding noise is suppressed.
Numerical Tips & Edge Cases
- Degenerate data: Three collinear points do not define a circle; identical points reduce rank. The solver will warn you.
- Scaling: Very large or very small inputs may be shown in scientific notation to avoid overflow and improve copyability.
- Parabola fit: For the “three-point vertical/horizontal” options,
x(ory) values must be distinct to solve the system. - Orientation: For rotated ellipses/hyperbolas, we display helper definitions for
x′andy′alongside the standard form.
Educational use: great for coordinate geometry, analytic geometry coursework, SAT/GCSE/A-level revision, and quick checks in CAD or simulation work. Everything runs client-side for speed and privacy.