Conic Sections Calculator - Identify the Conic Type

A conic sections calculator classifies the general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 as an ellipse, parabola, or hyperbola and reports the center, semi-axes, and eccentricity from the same six coefficients.

• • Classify a homework conic: type the A through F coefficients and read the type, eccentricity, and key axis values without computing the discriminant by hand.
• • Translate a shifted ellipse or hyperbola: feed a general equation with nonzero D and E to read the center (h, k) and the semi-axes a, b in the same pass.
• • Cross-check a fitted conic: compare a numerical conic fit from a measurement or regression against the closed-form (h, k) and eccentricity for the same equation.
• • Work a parabola from standard form: use the C=0 branch to read the vertex, focus, and directrix of a vertical or horizontal parabola written as y = ax^2 + bx + c or x = ay^2 + by + c.

Every conic section is the intersection of a plane with a double cone, and the same three families cover all of them: ellipses (including circles), parabolas, and hyperbolas. The calculator groups all three on one page so a shifted hyperbola and a standard parabola can be classified without switching tools.

When the input is a single parabola in standard form, the Parabola Calculator reads the vertex, focus, and directrix without expanding the equation into general form.

• A, B, C: Quadratic coefficients. The sign of B^2 - 4AC decides the family: positive for a hyperbola, zero for a parabola, negative for an ellipse.
• D, E: Linear coefficients. They hide the (h, k) shift inside the general equation, so the center has to be solved for.
• F: Constant term. Once the (h, k) shift is removed, F fixes the scale of the conic.
• e: Eccentricity. Zero for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, greater than 1 for a hyperbola.

For an ellipse, the conic discriminant is negative, so 4AC - B^2 is positive and the center formula divides cleanly. For a hyperbola, the discriminant is positive and the same shift gives a transverse axis a and conjugate axis b.

According to Wolfram MathWorld, a conic section is the curve formed by intersecting a plane with a double cone, and the general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 is classified by the discriminant B^2 - 4AC.

Once the conic is classified as an ellipse and the semi-axes a, b are known, the enclosed area follows from pi * a * b and the Ellipse Area Calculator returns the same number in a single read.

Four ideas show up every time a conic problem is solved by hand or by tool.

The conic discriminant B^2 - 4AC is the same kind of discriminant that classifies quadratic roots, and the Quadratic Formula Calculator handles the one-variable version of that classification.

The fastest path through this conic sections calculator is to drop in the six coefficients of the general form and read the type, center, and eccentricity off the primary result.

1. 1 Write the equation in general form: Expand the equation so it reads Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. For a parabola in y = ax^2 + bx + c, fill in A = a, B = 0, C = 0, D = b, E = -1, F = c.
2. 2 Enter A, B, C, D, E, F: Type the six coefficients into the input grid. Leave B = 0 for axis-aligned conics; set B nonzero to model a rotated ellipse or hyperbola.
3. 3 Read the type from the primary result: The first result row labels the curve as ellipse, parabola, or hyperbola. The discriminant row shows B^2 - 4AC, and the eccentricity row shows e in [0, 1) for ellipses, exactly 1 for a parabola, and above 1 for a hyperbola.
4. 4 Use the center and axis values to plot or check: The center (h, k), semi-axes a, b, and focal distance c are reported in the same units as the input coefficients. For a parabola, the same tool reports the vertex, focus, and directrix.
5. 5 Check the discriminant sign when the answer looks wrong: If the type does not match the textbook answer, the first place to look is the sign of B^2 - 4AC. A sign flip usually means a coefficient was entered with the wrong sign.

For 4x^2 + 9y^2 - 8x - 18y - 11 = 0, enter A = 4, B = 0, C = 9, D = -8, E = -18, F = -11 to get an ellipse with center (1, 1), a = 2.4495, b = 1.633, and eccentricity 0.7454.

If the conic turns out to be an axis-aligned ellipse and only the (h, k) center is needed, the Center Of Ellipse Calculator takes the same general form and returns the center with the supporting semi-axes and area.

A single conic sections tool covers the three families in one read.

• • Three families, one input grid: ellipse, parabola, and hyperbola share the same six coefficients, so swapping the input between a circle and a hyperbola takes only a sign change on A, C, or F. The discriminant handles the rest.
• • Real conic guards: the calculator refuses inputs that collapse to the empty set, a single point, a pair of lines, or a linear equation, so the center and eccentricity are never reported for a degenerate case.
• • Parabola fallback: when C = 0 or A = 0 the tool switches to the standard-form branch and reports the vertex, focus, and directrix instead of leaving the result blank.
• • Rotated conics work too: nonzero B in the input triggers the eigenvalue-based branch, so a rotated ellipse or hyperbola is handled the same way as an axis-aligned one and the angle of the major or transverse axis is reported.

If you are working through a problem set, the calculator lets you spend time on the classification step and treat the center and eccentricity as a sanity check rather than a re-derivation.

When a and b collapse to the same value the conic is a circle with e = 0, and the Circle Calculator runs the same center-plus-area calculation in the simpler pi * r^2 form.

Three to five things change the conic type and the read-out, and a couple are easy to overlook when typing the coefficients.

• • Degenerate inputs such as x^2 + y^2 = 0 (single point) or x^2 - y^2 = 0 (pair of intersecting lines) are not real conics and are rejected with a validation error rather than a misleading center or eccentricity.
• • The general-form branch requires both A and C to be nonzero; a parabola written as y = ax^2 + bx + c or x = ay^2 + by + c is handled by the parabola fallback rather than the discriminant branch.

If the type does not match the textbook answer, check the sign of the discriminant first, then the sign of F. A sign flip on either is the most common source of a wrong classification.

According to Wikipedia, Conic section article, the eccentricity e classifies the conic uniquely: e = 0 for a circle, 0 < e < 1 for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola.

For a conic with discriminant B^2 - 4AC negative and equal semi-axes, the Circle Equation Calculator reads the (h, k) center and radius r in the same form the general conic reduces to.