Conic section calculator
A conic section is any curve you get by slicing a cone with a flat plane: a circle, an ellipse, a parabola or a hyperbola. Every one of them can be written as a single second-degree equation in two variables, A x squared plus B x y plus C y squared plus D x plus E y plus F equals zero. The remarkable fact is that the type of curve is decided entirely by the three leading coefficients A, B and C, through a single number called the discriminant, B squared minus 4 A C. This calculator takes those coefficients and reports the discriminant and the resulting classification. A negative discriminant gives an ellipse, a discriminant of zero gives a parabola, and a positive discriminant gives a hyperbola, with a circle as the special case where A equals C and B is zero. The lower-order terms D, E and F shift and rotate the curve but never change its fundamental type. Enter your own coefficients to classify a curve or check an exercise. Every figure here is computed deterministically from the standard discriminant test, shown in full below with a worked example that reconciles exactly to the calculator.
The conic type is set by the discriminant B squared minus 4AC: negative is an ellipse, zero is a parabola, positive is a hyperbola. For A = 1, B = 0, C = 4 the discriminant is -16.00, so the curve is an ellipse.
Conic discriminant test
General conic: A x^2 + B x y + C y^2 + D x + E y + F = 0
Discriminant = B^2 - 4 A C
B^2 - 4AC < 0 -> ellipse (circle if A = C and B = 0)
B^2 - 4AC = 0 -> parabola
B^2 - 4AC > 0 -> hyperbola
Only the second-degree coefficients A, B and C control the shape of the curve. The discriminant B squared minus 4 A C is invariant under rotation and translation of the axes, so the linear terms D, E and F and the constant never change the classification.
Worked example
Classify the curve x squared plus 4 y squared equals 1, which has A = 1, B = 0, C = 4.
- B squared = 0 squared = 0
- 4 A C = 4 x 1 x 4 = 16
- Discriminant = 0 - 16 = -16
- The discriminant is negative, so the conic is an ellipse
- Since A does not equal C, it is a non-circular ellipse
The discriminant is -16.00 and the curve is an ellipse. These are the calculator's default inputs, so the result above matches the widget exactly.
Discriminant and conic type
| B squared minus 4AC | Conic type |
|---|---|
| Less than 0 | Ellipse (circle if A = C, B = 0) |
| Equal to 0 | Parabola |
| Greater than 0 | Hyperbola |
Degenerate cases (single point, line pair or no real curve) can occur for special constants.
Conic section calculator: frequently asked questions
What is a conic section?
A conic section is a curve formed by the intersection of a plane and a double cone. Depending on the angle of the cut you get a circle, an ellipse, a parabola or a hyperbola. Each can be written as a single second-degree polynomial equation in x and y, which is what this calculator classifies.
What is the discriminant of a conic?
For the general second-degree equation A x squared plus B x y plus C y squared plus lower terms, the discriminant is B squared minus 4 A C. Its sign determines the type of conic: negative for an ellipse, zero for a parabola and positive for a hyperbola. It is unchanged by rotating or shifting the axes.
How is a circle different from an ellipse?
A circle is a special ellipse where the two axes are equal. In coefficient terms, a circle requires A equal to C and B equal to zero, with a negative discriminant. This calculator flags that special case when it applies; otherwise a negative discriminant is reported as a general ellipse.
Why do D, E and F not appear in the test?
The coefficients D and E translate the curve and F sets its size or position, but none of them can turn an ellipse into a hyperbola or a parabola. The fundamental type is fixed by the quadratic part alone, captured by A, B and C, so only those enter the discriminant.
Can the result be degenerate?
Yes. For special combinations of all the coefficients a conic can collapse to a single point, a pair of lines, or have no real points at all. The discriminant test names the general type; checking for a degenerate case requires the full equation including the constant term. The arithmetic here is deterministic and matches the worked example exactly.
Official sources
- Mathematical functions and reference definitions: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.