# Conic Section Calculator - Foci, Eccentricity and Standard Form Online

> Free online conic section calculator: from Ax²+Bxy+Cy²+Dx+Ey+F=0 identify the curve and get the standard form, centre, foci, eccentricity and asymptotes.

- URL: https://qilingtools.com/en/math/conic
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Rearrange your equation into the general form and enter the six coefficients. The discriminant B²−4AC decides the type, completing the square gives the standard form, and the centre, vertices, foci, eccentricity, directrix and asymptotes are all worked out. A tilted curve with an xy term is rotated first.

## How to use

1. Rearrange the equation into Ax² + Bxy + Cy² + Dx + Ey + F = 0
2. Enter the six coefficients A to F (use 0 for missing terms)
3. Read the type, the standard form and every geometric quantity
4. Click “Plot this curve” to see the shape in the grapher

## Features

- The discriminant B²−4AC identifies a circle, ellipse, hyperbola or parabola
- Completing the square gives the standard form and the centre (or vertex)
- Ellipse: semi-axes, focal distance, vertices, foci, eccentricity, directrices, latus rectum and area
- Hyperbola: real and imaginary semi-axes, foci, eccentricity, asymptote equations and focal length
- Parabola: vertex, direction of opening, focus, directrix, focal parameter and latus rectum
- An xy term is handled by computing the rotation angle and reporting everything in the rotated frame
- Degenerate cases (a single point, the empty set, two lines) are named instead of being forced into a formula

## FAQ

### How do I tell an ellipse from a hyperbola or parabola?

Look at the discriminant B²−4AC: negative is the elliptic case (a circle when A=C and B=0), zero is parabolic and positive is hyperbolic. The value is listed for you.

### How is the eccentricity computed?

For both ellipses and hyperbolas e = c/a, where c is the focal distance and a is the semi-major (or real) axis. An ellipse has e between 0 and 1, a hyperbola has e greater than 1, and a parabola always has e = 1.

### What are the asymptotes of a hyperbola?

For a hyperbola centred at (h, k) with its real axis along x, they are y − k = ±(b/a)(x − h). The tool works out the actual slope.

### What if my equation has an xy term?

The curve is tilted. The tool computes the rotation angle θ = ½·arctan(B/(A−C)), turns the axes by it to remove the xy term, and reports the geometry in that rotated frame — so a rectangular hyperbola like xy = 1 is identified correctly.

### Why does it sometimes say “two straight lines”?

When the equation factors into two linear factors the graph is not a real conic but a pair of lines (x²−y²=0, for instance). Likewise x²+y²=0 is just the origin and x²+y²=−1 has no real graph at all.

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