Conic section

In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

Types of conics

Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an open curve and the plane is not parallel to a generator line of the cone, the figure is a hyperbola. (In this case the plane will intersect both halves of the cone, producing two separate curves, though often one is ignored.)

Related Topics:
Circle - Ellipse - Closed curve - Parabola - Hyperbola

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The degenerate cases, where the plane passes through the of the cone, resulting in an intersection figure of a point, a straight line, or a pair of lines, are often excluded from the list of conic sections.

Related Topics:
Point - Straight line

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In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. If the equation is of the form

Related Topics:
Cartesian coordinate system - Quadratic equation

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:ax^2 + 2hxy + by^2 +2gx + 2fy + c = 0;

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then:

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• if h² = ab, the equation represents a parabola;
• if h² < ab and a $e$ b and/or h$e$0 , the equation represents an ellipse;
• if h² > ab, the equation represents a hyperbola;
• if h² < ab and a = b and h = 0, the equation represents a circle;
• if a + b = 0, the equation represents a rectangular hyperbola.

Eccentricity

An alternative definition of conic sections starts with a point F (the focus), a line L (the directrix) not containing F and a positive number e (the eccentricity). The corresponding conic section consists of all points whose distance to F equals e times their distance to L. For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.

Related Topics:
Focus - Eccentricity

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For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is {a}over{e}, where a is the semi-major axis of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is ae.

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In the case of a circle e = 0 and one imagines the directrix to be infinitely far removed from the center. However, the statement that the circle consists of all points whose distance is e times the distance to L is not useful, because we get zero times infinity.

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The eccentricity of a conic section is thus a measure of how far it deviates from being circular.

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For a given a, the closer e is to 1, the smaller is the semi-minor axis.

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