Find circle, parabola, ellipse, hyperbola, and general conic equations from points, foci, directrix, parameters, or an existing equation in standard or general form. Private by design: calculations run locally in your browser.
Inputs
Results
Standard form
General quadratic formAx² + Bxy + Cy² + Dx + Ey + F = 0
Key features
Solution steps
Preview
Drag to pan. Use the mouse wheel or controls to zoom.
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
Circle through three points
Inputs: (0,3), (4,0), (1,-2)
Output: circle with center (1.045, 1.182)
The solver uses perpendicular bisectors, then reports radius, intercepts, area, and circumference.
Parabola from focus and directrix
Inputs: focus (2,1), directrix y = 0.5
Output: rotated/local standard form x'² = 1 y'
The vertex is halfway between the focus and directrix along the perpendicular axis.
Five rows solve the nullspace of Ax² + Bxy + Cy² + Dx + Ey + F = 0.
Formulas and methods
Case
Form or test
Required input
Circle
(x − h)² + (y − k)² = r²
Center and radius, diameter endpoints, or three non-collinear points.
Ellipse
x'²/a² + y'²/b² = 1
Foci and sum 2a, or center, axes, and rotation.
Hyperbola
x'²/a² − y'²/b² = 1
Foci and difference 2a, or center, axes, branch direction, and rotation.
Parabola
x'² = 4py'
Focus/directrix, vertex/point/axis, or three points for an axis-aligned fit.
General conic
B² − 4AC classifies the quadratic part.
Five points or pasted Ax² + Bxy + Cy² + Dx + Ey + F = 0.
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 formAx² + 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.
FAQ
How many points define a conic?
Five points usually define a unique conic because the general equation has six coefficients and one overall scale factor. The points must not make a rank-deficient or degenerate system.
Can this find a conic from 5 points?
Yes. Choose the 5 points mode, enter P1 through P5, and the calculator solves Ax² + Bxy + Cy² + Dx + Ey + F = 0, classifies the conic, and converts to standard form when possible.
How do I convert general form to standard form?
Choose Analyze, paste an equation such as x^2 + y^2 - 4x + 2y - 11 = 0, and compute. The tool completes the square or rotates axes for Bxy terms.
How do I know if the conic is a circle, ellipse, parabola, or hyperbola?
The main test is B² − 4AC. Negative gives an ellipse or circle, zero gives a parabola, and positive gives a hyperbola, with extra checks for degeneracy and imaginary cases.
Why does my input have no solution?
The data may be repeated, collinear in a way that lowers rank, inconsistent with the selected method, or an equation that represents lines, a point, or no real curve.
Does Bxy mean the conic is rotated?
For ordinary conics, a nonzero Bxy term means the principal axes are rotated relative to the x and y axes. The analyzer reports the rotation angle and local coordinates.
Are calculations done locally?
Yes. The page runs the math in your browser and does not upload your inputs.
