Conic section calculator

This conic section calculator identifies the curve behind Ax² + Bxy + Cy² + Dx + Ey + F = 0 – circle, ellipse, parabola, hyperbola or a degenerate case – and works out its standard form, center, axes, foci, vertices, directrix, asymptotes and eccentricity. Example: 4x² + 9y² − 16x + 18y − 11 = 0 is an ellipse with center (2, −1).

Example: A (x²) 4 · B (xy) 0 · C (y²) 9 · D (x) -16 · E (y) 18 · F (constant) -11 → Type: This conic section is an ellipse.. Source: Wikipedia, Conic section: discriminant B² − 4AC, 3 × 3 determinant for degenerate conics, eccentricity in terms of coefficients (read 15.10.2026). Updated: .

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Result

Type
This conic section is an ellipse.
Properties
PropertyValue
Typean ellipse
Discriminant B² − 4AC−144
Determinant of the 3 × 3 matrix−1,296
Standard form(x − 2)²/9 + (y + 1)²/4 = 1
Center(2, −1)
Semi-major axis a3
Semi-minor axis b2
Eccentricity e0.745356
Foci(−0.236068, −1); (4.236068, −1)
Vertices(−1, −1); (5, −1)
Calculated from the discriminant B² − 4AC and the 3 × 3 determinant (Wikipedia, Conic section); position and axes by rotating to principal axes. Values are rounded to 7 significant digits.

How it is calculated

How to identify a conic section

Every conic section can be written as Ax² + Bxy + Cy² + Dx + Ey + F = 0. The discriminant B² − 4AC decides the type: below 0 an ellipse (a circle if A = C and B = 0), exactly 0 a parabola, above 0 a hyperbola (a rectangular one if A + C = 0). For 4x² + 9y² − 16x + 18y − 11 = 0 the discriminant is 0 − 4 · 4 · 9 = −144, so the curve is an ellipse.

ABCDEFB² − 4ACType
409−1618−11−144an ellipse
101−46−12−4a circle
001−8−2170a parabola
10−4−20−1516a hyperbola
01000−11a rectangular hyperbola
10−10004two intersecting lines (degenerate hyperbola)

Degenerate conics

The discriminant alone cannot tell a real curve from a degenerate one. For that the calculator uses the determinant of the 3 × 3 matrix, (AC − B²/4)F + (BED − CD² − AE²)/4. If it is 0, the “curve” is a single point, two crossing lines (x² − y² = 0 gives y = x and y = −x) or two parallel lines. If an ellipse equation has no real solutions at all, such as x² + y² + 1 = 0, the result is empty.

From the general form to the standard form

Completing the square turns the example into 4(x − 2)² + 9(y + 1)² = 36, that is (x − 2)²/9 + (y + 1)²/4 = 1: center (2, −1), semi-axes a = 3 and b = 2, foci at a distance c = √(9 − 4) = 2.236068 from the center, eccentricity e = c/a = 0.745356. The parabola y² − 2y − 8x + 17 = 0 becomes (y − 1)² = 8(x − 2): vertex (2, 1), p = 2, focus (4, 1), directrix x = 0. With an xy term (B ≠ 0) the axes are turned by the angle θ with tan 2θ = B/(A − C); xy = 1 is a hyperbola turned by 45°.

Eccentricity

Frequently asked questions

How do you identify a conic section from its equation?

Compute B² − 4AC. Negative means ellipse (circle when A = C and B = 0), zero means parabola, positive means hyperbola – unless the conic is degenerate.

What type of conic is 4x² + 9y² − 16x + 18y − 11 = 0?

An ellipse: B² − 4AC = −144. Its standard form is (x − 2)²/9 + (y + 1)²/4 = 1 with center (2, −1).

What is a degenerate conic?

A conic whose 3 × 3 determinant is 0: a single point, two intersecting lines or two parallel (possibly coinciding) lines. Example: x² − y² = 0 gives the lines y = x and y = −x.

What does the xy term do?

It rotates the curve. The axes are turned by θ with tan 2θ = B/(A − C); for xy = 1 the angle is 45°.

What is the eccentricity of a parabola?

Exactly 1. Circles have 0, ellipses between 0 and 1 and hyperbolas more than 1.

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