Conic Sections Calculator
Solve conic sections problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Conic Sections Calculator
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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.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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