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iam weak in conicsections.how to prepare for that
Dear Sreekar. Conic sections are the curves which can be derived from taking slices of a "double-napped" cone. (A double-napped cone, in regular English, is two cones "nose to nose", with the one cone balanced perfectly on the other.) "Section" here is used in a sense similar to that in medicine or science, where a sample (from a biopsy, for instance) is frozen or suffused with a hardening resin, and then extremely thin slices ("sections") are shaved off for viewing under a microscope. If you think of the double-napped cones as being hollow, the curves we refer to as conic sections are what results when you section the cones at various angles. Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved
Dear Sreekar.
Conic sections are the curves which can be derived from taking slices of a "double-napped" cone. (A double-napped cone, in regular English, is two cones "nose to nose", with the one cone balanced perfectly on the other.) "Section" here is used in a sense similar to that in medicine or science, where a sample (from a biopsy, for instance) is frozen or suffused with a hardening resin, and then extremely thin slices ("sections") are shaved off for viewing under a microscope. If you think of the double-napped cones as being hollow, the curves we refer to as conic sections are what results when you section the cones at various angles. Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved
There are plenty of sites and books with pictures illustrating how to obtain the various curves through sectioning, so I won''t bore you with more pictures here. And there are books and entire web sites devoted to the history of conics, the derivation and proofs of their formulas, and their various applications. I will not attempt to reproduce that information here.
This lesson, and the conic-specific lessons to which this page links, will instead concentrate on: finding curves, given points and other details; finding points and other details, given curves; and setting up and solving conics equations to solve typical word problems.
There are some basic terms that you should know for this topic:
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In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity. Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well. The conic sections were named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity.
Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well.
The conic sections were named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
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