Conic Sections Calculator
Calculate properties of conic sections including ellipses, circles, parabolas, and hyperbolas. Find foci, vertices, eccentricity, and equations.
Conic Sections Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: Ellipse: x^2/a^2 + y^2/b^2 = 1; Parabola: y = a(x-h)^2 + k; Hyperbola: x^2/a^2 - y^2/b^2 = 1
Additional inputs: Vertex k, Coefficient a, Center h, Center k.
Worked example — Foci: (-2,-1), (6,-1) | e = 0.8 | Area = 47.12
Formula
Ellipse: x^2/a^2 + y^2/b^2 = 1; Parabola: y = a(x-h)^2 + k; Hyperbola: x^2/a^2 - y^2/b^2 = 1
Each conic section has a standard form equation. The parameters a, b determine the shape while h, k define the center or vertex. The relationship between a, b, and c (focal distance) differs: for ellipses c^2 = a^2 - b^2, for hyperbolas c^2 = a^2 + b^2.
Worked Examples
Example 1: Ellipse Properties
Problem:Find the foci, eccentricity, and area of the ellipse (x-2)^2/25 + (y+1)^2/9 = 1.
Solution:Center: (2, -1), a = 5 (horizontal), b = 3 c = sqrt(25 - 9) = sqrt(16) = 4 Foci: (2-4, -1) = (-2, -1) and (2+4, -1) = (6, -1) Eccentricity: e = c/a = 4/5 = 0.8 Area = pi x 5 x 3 = 15pi = 47.124 Perimeter (Ramanujan) = pi(3(5+3) - sqrt((15+3)(5+9))) = 25.527
Result:Foci: (-2,-1), (6,-1) | e = 0.8 | Area = 47.12
Example 2: Parabola Focus and Directrix
Problem:Find the focus, directrix, and latus rectum of y = 0.5(x-3)^2 + 2.
Solution:Vertex: (3, 2), a = 0.5 (opens upward) Focal length = 1/(4|a|) = 1/(4 x 0.5) = 0.5 Focus: (3, 2 + 0.5) = (3, 2.5) Directrix: y = 2 - 0.5 = 1.5 Latus rectum = 4 x 0.5 = 2 Axis of symmetry: x = 3
Result:Focus: (3, 2.5) | Directrix: y = 1.5 | Latus Rectum: 2
Frequently Asked Questions
What are conic sections and how are they formed?
Conic sections are curves obtained by intersecting a double-napped cone with a plane at different angles. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas. When the cutting plane is perpendicular to the axis of the cone, the intersection is a circle. When the plane is tilted but still intersects only one nappe of the cone, the result is an ellipse. When the plane is parallel to the slant of the cone, the intersection is a parabola. When the plane intersects both nappes, the result is a hyperbola. Conic sections were first studied by the ancient Greek mathematician Apollonius of Perga around 200 BCE and have been fundamental to mathematics, physics, and engineering ever since. They describe planetary orbits, satellite trajectories, and reflective surface properties.
What is the eccentricity of a conic section and what does it tell us?
Eccentricity is a non-negative number that describes how much a conic section deviates from being circular. It is typically denoted by the letter e. A circle has eccentricity exactly equal to 0, meaning it has no deviation from circularity. An ellipse has eccentricity between 0 and 1, with values closer to 0 being more circular and values closer to 1 being more elongated. A parabola has eccentricity exactly equal to 1. A hyperbola has eccentricity greater than 1, with larger values indicating wider opening branches. For ellipses, eccentricity is calculated as e equals c divided by a, where c is the distance from center to focus and a is the semi-major axis length. Earth orbit around the Sun has an eccentricity of about 0.017, making it nearly circular.
What are the practical applications of parabolas?
Parabolas have a remarkable reflective property: any ray parallel to the axis of symmetry reflects off the parabolic surface and passes through the focus, and vice versa. This property is exploited in satellite dishes, radio telescopes, car headlights, flashlights, and solar concentrators. Satellite dishes are parabolic so that incoming signals concentrate at the focal point where the receiver is placed. Headlights place the bulb at the focus so light reflects outward in a parallel beam. The trajectory of a projectile in uniform gravity, ignoring air resistance, follows a parabolic path. Suspension bridge cables approximate parabolic shapes under uniform load. Parabolic microphones used in sports broadcasting collect distant sounds by focusing them at the microphone placed at the focus point of the dish.
How do I determine the type of conic from a general second-degree equation?
The general second-degree equation is Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equals 0. The discriminant B squared minus 4AC determines the conic type. If the discriminant is negative and A equals C with B equal to 0, the conic is a circle. If the discriminant is negative but A does not equal C, it is an ellipse. If the discriminant equals 0, it is a parabola. If the discriminant is positive, it is a hyperbola. Note that degenerate cases exist where the equation represents a point, line, or pair of lines rather than a proper conic. To convert from general form to standard form, you typically complete the square for both x and y terms, which reveals the center, radii, or other key parameters of the conic section.
What is the relationship between conic sections and orbital mechanics?
Johannes Kepler discovered that planetary orbits are ellipses with the Sun at one focus, which became his first law of planetary motion published in 1609. Isaac Newton later proved mathematically that any two-body gravitational system produces orbits that follow conic sections. The specific conic type depends on the total energy of the orbiting body. Negative total energy produces an elliptical orbit, which is a bound orbit like planets and moons. Zero total energy produces a parabolic trajectory, representing the minimum escape velocity. Positive total energy produces a hyperbolic trajectory, like interstellar objects passing through our solar system. Comets can follow any conic path: periodic comets have elliptical orbits while one-time visitors follow parabolic or hyperbolic paths. This relationship between gravity and conic geometry is fundamental to spacecraft trajectory planning.
What are the asymptotes of a hyperbola and how do you find them?
Asymptotes are the two straight lines the hyperbola's branches approach without ever touching as x grows large. For a horizontal hyperbola (x-h)^2/a^2 - (y-k)^2/b^2 = 1 they are y = k plus or minus (b/a)(x - h); for a vertical hyperbola the slopes become plus or minus a/b. Geometrically, draw the central rectangle of width 2a and height 2b centred at (h, k): the asymptotes are its extended diagonals. They are the fastest way to sketch a hyperbola accurately, because the vertices and the asymptotes together pin down both branches.
What is the latus rectum and why is it reported?
The latus rectum is the chord that passes through a focus perpendicular to the principal axis of the conic, and its length is a compact measure of how wide the curve is at the focus. For an ellipse or a hyperbola it equals 2b^2/a, so the ellipse with a = 5 and b = 3 has a latus rectum of 18/5 = 3.6. For a parabola it equals 4p, where p is the focal length, so a parabola with focal length 0.5 has a latus rectum of 2. Optical and antenna designers use it to size a reflector's aperture relative to its focus.
How do the focus and directrix define every conic at once?
Every conic is the set of points whose distance to a fixed focus divided by its distance to a fixed line, the directrix, is a constant equal to the eccentricity e. That single definition generates all the types: e less than 1 gives an ellipse, e exactly 1 gives a parabola, where the two distances are always equal, and e greater than 1 gives a hyperbola. A circle is the limiting case where the directrix retreats to infinity. This is why a parabola's focus and directrix appear together in its standard output: they are the two objects the curve is balanced between.
Why is c^2 = a^2 - b^2 for an ellipse but c^2 = a^2 + b^2 for a hyperbola?
The sign follows from the defining focal property of each curve. An ellipse is the locus where the sum of the distances to the two foci equals 2a, which forces the foci inside the curve and makes c smaller than a, so c^2 = a^2 - b^2 and e = c/a stays below 1. A hyperbola is the locus where the absolute difference of those distances equals 2a, which places the foci beyond the vertices, so c exceeds a and c^2 = a^2 + b^2, giving e above 1. For the hyperbola b is the conjugate semi-axis and sets the asymptote slope rather than a physical width.
What is a degenerate conic and when does one appear?
A degenerate conic arises when the cutting plane passes through the apex of the double cone, so the intersection collapses. Depending on the tilt you get a single point, one line, or a pair of intersecting lines instead of a proper curve. Algebraically these still satisfy the general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, so the B^2 - 4AC test alone cannot rule them out. A telltale sign after completing the square is a right-hand side of zero, as in x^2 - 4y^2 = 0, which factors into two crossing lines rather than a hyperbola.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.
🧮Area of Crescent Calculator
Calculate area of crescent with inputs, formulas, and instant results.
🧮Center of Mass Calculator
Calculate center of mass with inputs, formulas, and instant results.
🧮Centroid Calculator
Calculate centroid with inputs, formulas, and instant results.
🧮Chord Length Calculator
Calculate chord length with inputs, formulas, and instant results.
🧮Cross Sectional Area Calculator
Calculate cross sectional area with inputs, formulas, and instant results.