QiLing Tools

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

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.

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How to use Conic Sections

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

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.