Conic Equation Finder Calculator

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.
• General conic — from five points, or by analyzing a pasted equation such as 9x² + 4y² = 36.

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.

Worked examples

Inputs: (0,3), (4,0), (1,-2)
Output: circle with center (1.045, 1.182)

Inputs: focus (2,1), directrix y = 0.5
Output: rotated/local standard form x'² = 1 y'

Inputs: foci (-2,0), (2,0), sum 6
Output: x²/9 + y²/5 = 1

Inputs: center (0,0), a=4, b=3
Output: x²/16 − y²/9 = 1

Input: 9x² + 4y² = 36
Output: x²/4 + y²/9 = 1, ellipse

Inputs: (-3,0), (3,0), (0,2), (0,-2), (2,1.491)
Output: conic classified as ellipse

Formulas and methods

Common failure cases include repeated points, rank-deficient five-point systems, a focus on its directrix, invalid axis lengths, or equations that reduce to lines, a point, or no real graph.

Accuracy and limitations

Calculations run client-side. Coefficients are normalized for readable output, and tiny floating-point noise is rounded when displayed. The unrounded values are used for the graph and derived properties.

The analyzer uses the quadratic discriminant B² − 4AC, the determinant of the conic matrix, completing the square, and eigen-axis rotation for Bxy terms. It flags degenerate, imaginary, and rank-deficient inputs when the numeric tests indicate no ordinary real conic.

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 (or y) values must be distinct to solve the system.
• Orientation: For rotated ellipses/hyperbolas, we display helper definitions for x′ and y′ 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.