Ellipse (geometry)
An ellipse is a closed conic section defined as the set of all points in a plane for which the sum of the distances to two fixed points (called foci) is constant. It resembles a flattened circle and is characterised by its semi-major axis a (longest radius) and semi-minor axis b (shortest radius), with eccentricity e = c/a where c = √(a² − b²) gives the focal distance. Ellipses appear throughout nature and physics, most famously as the shapes of planetary orbits as described by Kepler's first law.
Key Formula
x²/a² + y²/b² = 1
LaTeX: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1
| Symbol | Meaning | Unit |
|---|---|---|
| a | Semi-major axis (half the longest diameter) | m or unitless |
| b | Semi-minor axis (half the shortest diameter) | m or unitless |
| c | Focal distance from centre; c = √(a²−b²) | m or unitless |
| e | Eccentricity; e = c/a (0 ≤ e < 1) | unitless |
Worked Example
Problem
An ellipse has equation x²/25 + y²/9 = 1. Find the semi-major axis, semi-minor axis, foci, and eccentricity.
Solution
Step 1 — Read axes: a² = 25, b² = 9, so a = 5, b = 3. Step 2 — Focal distance: c = √(a²−b²) = √(25−9) = √16 = 4. Step 3 — Foci: (±4, 0). Step 4 — Eccentricity: e = c/a = 4/5 = 0.8.
Answer
Semi-major a = 5, semi-minor b = 3, foci (±4, 0), eccentricity e = 0.8
Properties of an Ellipse (x²/a² + y²/b² = 1, a > b)
| Property | Formula | Description | Example (a=5, b=3) |
|---|---|---|---|
| Semi-major axis | a | Half the longest diameter | a = 5 |
| Semi-minor axis | b | Half the shortest diameter | b = 3 |
| Focal distance | c = √(a²−b²) | Centre to each focus | c = 4 |
| Eccentricity | e = c/a | 0 = circle, 1 = parabola limit | e = 0.8 |
| Area | A = πab | Enclosed area | A = 15π ≈ 47.12 |
| Perimeter (approx) | P ≈ 2π√((a²+b²)/2) | Ramanujan approximation | P ≈ 25.53 |
Interactive Tools
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Related Terms
Conic Section
A conic section is a curve obtained by intersecting a right circular cone with a plane at various angles, yielding four distinct types: circle, ellipse, parabola, and hyperbola. These curves were first studied systematically by the Greek mathematician Apollonius of Perga (c. 262–190 BCE) and later became fundamental in physics when Kepler showed that planetary orbits are ellipses. Conic sections are described by the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, and the type is determined by the discriminant B² − 4AC.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system in which points are represented by numerical coordinates. It provides a powerful bridge between algebra and geometry, allowing geometric shapes to be described by equations and algebraic operations to yield geometric insights. Key concepts include plotting points, finding distances, midpoints, slopes, and the equations of lines, circles, and conic sections on the Cartesian plane.
Midpoint Formula
The midpoint formula gives the coordinates of the point that lies exactly halfway between two given points on a line segment in the Cartesian plane. It is derived by averaging the x-coordinates and the y-coordinates of the two endpoints separately. The midpoint is used extensively in geometry proofs, construction of perpendicular bisectors, and in applications such as GPS interpolation and computer graphics.
From Greek "elleipsis" meaning "a falling short" or "deficiency" (from "elleipein", to fall short), named by Apollonius of Perga (c. 200 BCE) because the plane intersects the cone at an angle that "falls short" of producing a parabola. The term entered Latin as "ellipsis" and English mathematical usage in the 17th century.