Conic Section Calculator
Last updated: 27 August 2026
Reviewed by Gavin Meiring, Lead research and primary author · Doctoral Candidate (Corporate Governance) · Research and drafting assisted by AI
- The Greek mathematician Apollonius of Perga wrote an eight-book treatise called 'Conics' around 200 BCE, naming the ellipse, parabola, and hyperbola.
- A conic section is literally the curve produced by slicing a cone with a plane: shallow, parallel, and steep cuts give circles, ellipses, parabolas, and hyperbolas.
- Kepler's first law of 1609 states that planets orbit the Sun in ellipses with the Sun at one focus.
Conic Section Calculator
A conic section calculator analyses equations and properties of circles, ellipses, parabolas, and hyperbolas: the four curves formed by slicing a cone with a plane. It is used by A-level and university students studying coordinate geometry, engineers designing satellite dishes and lenses, and astronomers working with orbital mechanics.
How to Use the Conic Section Calculator
- Enter the conic section equation in standard form or general form (Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0).
- Select the conic type if known, or let the tool identify it automatically from the discriminant.
- Click Calculate to see the key properties: centre, vertices, foci, eccentricity, and axis lengths.
- View the plotted curve on a coordinate grid.
- Enter specific points to check whether they lie on the conic or to find tangent lines.
The Formula
The general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 defines a conic. The discriminant B^2 - 4AC determines the type:
B^2 - 4AC < 0: ellipse (including circle when A=C and B=0) B^2 - 4AC = 0: parabola B^2 - 4AC > 0: hyperbola
Standard forms (with axes aligned to coordinate axes): Circle: (x-h)^2 + (y-k)^2 = r^2 Ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1 Parabola: (y-k) = a(x-h)^2 or (x-h) = a(y-k)^2 Hyperbola: (x-h)^2/a^2 - (y-k)^2/b^2 = 1
Real-World Example
Identify and analyse the conic: x^2/25 + y^2/9 = 1
This is an ellipse in standard form with centre at the origin (h=0, k=0).
a^2 = 25, so a = 5 (semi-major axis along x-axis) b^2 = 9, so b = 3 (semi-minor axis along y-axis)
c^2 = a^2 - b^2 = 25 - 9 = 16, so c = 4 (distance from centre to each focus)
Foci: (-4, 0) and (4, 0) Vertices: (-5, 0), (5, 0), (0, -3), (0, 3) Eccentricity: e = c/a = 4/5 = 0.8
The ellipse is fairly elongated (e = 0.8 is close to 1, the limit of a parabola).
Conics in Science and Engineering
Conic sections appear throughout physics and engineering at every scale. Planetary orbits are ellipses, with the sun at one focus: this was Kepler's first law of planetary motion, published in 1609. Comets and space probes follow parabolic or hyperbolic paths when they pass near a massive body with enough speed to escape its gravity. Parabolic mirrors focus parallel light rays to a single point (the focus), which is why satellite dishes, radio telescopes, car headlights, and solar concentrators all use parabolic shapes. Elliptical mirrors and "whispering gallery" rooms reflect sound or light from one focus to the other, regardless of the path taken. Hyperbolas appear in navigation systems (LORAN and GPS triangulation) because the set of all points with a fixed difference in distances from two locations forms a hyperbola. Engineers designing lenses, antennas, and reflective surfaces rely on conic section geometry daily.
Frequently Asked Questions
What is eccentricity and what does it measure? Eccentricity (e) measures how far a conic departs from being a perfect circle. For a circle, e = 0. For an ellipse, 0 < e < 1. For a parabola, e = 1 exactly. For a hyperbola, e > 1. The closer e is to 0, the more nearly circular the ellipse. The closer e is to 1, the more elongated it becomes before transitioning to a parabola.
What are the foci of a conic section? Each conic (except a circle, which has a single centre) has one or two special points called foci. For an ellipse, the sum of distances from any point on the ellipse to the two foci is constant and equals 2a. For a parabola, every point is equidistant from the single focus and a line called the directrix. For a hyperbola, the difference (not sum) of distances from any point to the two foci is constant.
How do I convert from general form to standard form? Use the method of completing the square on both x and y terms. Group x-terms together, group y-terms together, add the appropriate constants to both sides to complete each square, then rewrite in the form (x-h)^2/a^2 + (y-k)^2/b^2 = 1 (or the equivalent for parabolas and hyperbolas). This reveals the centre, orientation, and axis lengths directly.
What is the difference between a parabola and a hyperbola? A parabola is the set of points equidistant from a focus and a directrix line; it has one branch and opens in one direction. A hyperbola is the set of points where the difference of distances to two foci is constant; it has two separate branches opening in opposite directions. Both have asymptotic behaviour at large distances from the centre, but the parabola has no asymptotes while the hyperbola approaches its asymptotes without touching them.
Also try these free tools:
A hyperbola worked end to end
Take x^2/16 minus y^2/9 = 1. The positive term carries the x, so the transverse axis lies along the x-axis.
a^2 is 16, so a is 4 and the vertices sit at (4, 0) and (minus 4, 0). b^2 is 9, so b is 3. For a hyperbola c^2 is a^2 plus b^2, not a^2 minus b^2 as it is for an ellipse, so c^2 is 16 plus 9, or 25, and c is 5. The foci sit at (5, 0) and (minus 5, 0).
Eccentricity is c divided by a, so 5 divided by 4, or 1.25. A figure above 1 describes a hyperbola, and 1.25 gives a moderately open pair of branches.
The asymptotes are the lines y = 0.75x and y = minus 0.75x. The semi-latus rectum, the length of the chord through a focus parallel to the conjugate axis, is 2b^2 divided by a, which is 18 divided by 4, or 4.5.
One substitution confirms the equation before you trust the rest. Take x = 5, the focus coordinate, and solve for y: y^2/9 = 25/16 minus 1 = 0.5625, so y^2 = 5.0625 and y = 2.25. Put the point back into the equation and 25/16 minus 5.0625/9 is 1.5625 minus 0.5625, or exactly 1. The point (5, 2.25) lies on the curve, where the arithmetic says it should.
The discriminant decides the type
Given the general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, the quantity B^2 minus 4AC names the curve before you complete a single square. A negative value gives an ellipse, zero gives a parabola, and a positive value gives a hyperbola.
| Equation | B^2 minus 4AC | Type |
|---|---|---|
| x^2 + y^2 - 4 = 0 | -4 | ellipse |
| 4x^2 + 9y^2 - 36 = 0 | -144 | ellipse |
| x^2 - 4y = 0 | 0 | parabola |
| x^2 - y^2 - 9 = 0 | 4 | hyperbola |
| xy = 1 | 1 | hyperbola |
| 2x^2 + 2xy + 2y^2 - 4 = 0 | -12 | ellipse |
Two rows repay a closer look. The equation xy = 1 carries no squared terms at all, yet the discriminant is 1 and the curve is the rectangular hyperbola. The equation 2x^2 + 2xy + 2y^2 minus 4 = 0 carries both squared terms and a cross term, and the discriminant of minus 12 makes it an ellipse rotated away from the coordinate axes.
The four conics side by side
| Curve | Standard form | Eccentricity | Asymptotes | Latus rectum |
|---|---|---|---|---|
| Circle | (x-h)^2 + (y-k)^2 = r^2 | 0 | none | 2r |
| Ellipse | (x-h)^2/a^2 + (y-k)^2/b^2 = 1 | between 0 and 1 | none | 2b^2/a |
| Parabola | y-k = a(x-h)^2 | 1 | none | 4p, where p is the focal distance |
| Hyperbola | (x-h)^2/a^2 minus (y-k)^2/b^2 = 1 | above 1 | y = plus or minus (b/a)x | 2b^2/a |
Three worked figures make the last column concrete. The ellipse x^2/25 + y^2/9 = 1 has c^2 = 25 minus 9 = 16, so c = 4, e = 0.8, and a latus rectum of 18/5, or 3.6. The rounder ellipse x^2/9 + y^2/4 = 1 has c^2 = 9 minus 4 = 5, so c = 2.236068, e = 0.745356, and a latus rectum of 8/3, or 2.666667. The parabola y = x^2/8 has 4p = 8, so p = 2, the focus sits at (0, 2), the directrix is the line y = minus 2, and the latus rectum is 8.0. Smaller eccentricity means a rounder curve, and the latus rectum grows with the curve's openness.
Method and assumptions behind the discriminant test
The standard forms in the table assume the axes of the curve run parallel to the coordinate axes. Four conditions keep the analysis valid.
- A cross term means the curve is rotated. When B is not zero, the axes of the conic do not line up with x and y, and the standard forms do not apply until you rotate the coordinate system or diagonalise the quadratic part.
- The equation must be genuinely second degree. If A, B and C are all zero the equation describes a line, and the discriminant test has nothing to test.
- Some equations with a valid discriminant are degenerate. x^2 plus y^2 equals 0 has a negative discriminant and describes a single point rather than a circle. x^2 minus y^2 equals 0 has a positive discriminant and describes a pair of lines rather than a hyperbola.
- From general form you must complete the square. The centre, the vertices and the axis lengths are readable only after the x and y terms are grouped and squared off, which the tool does for you and which every student should do once by hand.
A reader who knows those four conditions can tell which answers to trust. A reader who does not will accept a rotated ellipse described as an unrotated one, and the numbers will not match the sketch.
A note on the source
The curves carried these names long before they carried these equations. Apollonius of Perga wrote eight books on the subject, and his Conics, composed around 200 BC, treats the ellipse, the parabola and the hyperbola as sections of a single cone and derives their properties without coordinates. The coordinate treatment used above arrived much later, and the physics later still: Kepler published the finding that planetary orbits are ellipses with the sun at one focus in 1609.