Parabola: Difference between revisions
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[[Image:Parabola.jpg|thumb|right|400px|<small>Parabola y = 2''px''<sup>2</sup> (red line). Focus F is at (0,''p''/2). Distance FP is equal to distance PA. Blue dashed horizontal line is directrix | [[Image:Parabola.jpg|thumb|right|400px|<small>Parabola y = 2''px''<sup>2</sup> (red line). Focus F is at (0,''p''/2). Distance FP is equal to distance PA. Blue dashed horizontal line is directrix. </small> ]] | ||
A '''parabola''' is the planar [[curve]] formed by the points that lie as far from a given [[line (geometry)|line]] (the ''directrix'') as from a given [[point (geometry)|point]] (the ''focus''). Alternatively, a parabola is the curve you get when intersecting a right circular [[cone]] with a plane parallel to a generator of the cone (a line on the cone which goes through its apex); thus, a parabola is a [[conic section]]. | A '''parabola''' is the planar [[curve]] formed by the points that lie as far from a given [[line (geometry)|line]] (the ''directrix'') as from a given [[point (geometry)|point]] (the ''focus''). Alternatively, a parabola is the curve you get when intersecting a right circular [[cone]] with a plane parallel to a generator of the cone (a line on the cone which goes through its apex); thus, a parabola is a [[conic section]]. | ||
Revision as of 04:51, 18 March 2010
A parabola is the planar curve formed by the points that lie as far from a given line (the directrix) as from a given point (the focus). Alternatively, a parabola is the curve you get when intersecting a right circular cone with a plane parallel to a generator of the cone (a line on the cone which goes through its apex); thus, a parabola is a conic section.
If the focus lies on the directrix, then the "parabola" is in fact a line. In the language of conic sections, this corresponds to the case when the plane contains a generator of the cone.
To avoid this degenerate case, we assume that the focus lies not on the directrix. The line through the focus and perpendicular to the directrix is called the axis of the parabola. It is the unique line of symmetry of the parabola. The parabola has one point that lies on the axis. This point is called the vertex of the parabola. The distance between the focus and the vertex is called the focal distance of the parabola. It is the same as the distance between the vertex and the directrix, and half the distance from the focus to the directrix.
The shape of a parabola is determined by the focal distance. All parabola with the same focal distance are congruent, meaning that given any parabola can be moved to any other parabola with the same focal distance by a rigid motion.