Lecture

Conic Sections: Definitions and Equations

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Description

This lecture covers the general definition of a conic based on eccentricity, focusing on parabolas, ellipses, and hyperbolas. It explains the equations of these conic sections and their geometric properties, such as directrix and focus. The lecture also delves into applications of conic sections in geometry and provides examples to illustrate the concepts.

Official source

Ontological neighbourhood

Mathematics

Geometry: Euclidean geometry

Related concepts (32)

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Parabola

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus.

Confocal conic sections

In geometry, two conic sections are called confocal if they have the same foci. Because ellipses and hyperbolas have two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally (at right angles). Parabolas have only one focus, so, by convention, confocal parabolas have the same focus and the same axis of symmetry.

Conic section

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions.

Hyperbola

In mathematics, a hyperbola (haɪˈpɜrbələ; pl. hyperbolas or hyperbolae -liː; adj. hyperbolic ˌhaɪpərˈbɒlɪk) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone.

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