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.
Key Formula
Ax² + Bxy + Cy² + Dx + Ey + F = 0
LaTeX: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
| Symbol | Meaning | Unit |
|---|---|---|
| A, B, C, D, E, F | Real constant coefficients | unitless |
| B²−4AC | Discriminant determining conic type | unitless |
Worked Example
Problem
Classify the conic: 4x² + 9y² = 36. State its type and find its semi-axes.
Solution
Step 1 — Rewrite in standard form: x²/9 + y²/4 = 1. Step 2 — Compare with x²/a² + y²/b² = 1: a² = 9, b² = 4, so a = 3, b = 2. Step 3 — Since a ≠ b and both terms positive, this is an ellipse with semi-major axis a = 3 and semi-minor axis b = 2.
Answer
Ellipse; semi-major axis a = 3, semi-minor axis b = 2
Classification of Conic Sections by Discriminant and Shape
| Conic Type | Discriminant B²−4AC | Standard Form | Example Equation |
|---|---|---|---|
| Circle | < 0, A = C, B = 0 | x² + y² = r² | x² + y² = 25 |
| Ellipse | < 0 (general) | x²/a² + y²/b² = 1 | 4x²+9y²=36 |
| Parabola | = 0 | y = ax² or x = ay² | y = 2x² |
| Hyperbola | > 0 | x²/a² − y²/b² = 1 | x²/9 − y²/4 = 1 |
Interactive Tools
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Related Terms
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.
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.
Distance Formula
The distance formula gives the straight-line (Euclidean) distance between two points in the Cartesian plane using their coordinates. It is derived directly from the Pythagorean theorem by treating the horizontal and vertical separations as legs of a right triangle. The formula extends naturally to three dimensions as d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) and is fundamental in analytic geometry, physics, and data science.
From Greek "konikos" (of or like a cone), derived from "konos" (cone). The term "conic section" was introduced by Apollonius of Perga in his landmark eight-volume work "Conics" (c. 200 BCE), which classified all possible plane intersections of a double cone.