A parabola
In mathematics, the parabola (plural parabolae or parabolas, from the Greek παραβολή) is a conic section, the intersection
of a right circular conical surface and a plane parallel to a generating straight line of that surface. Given a
point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistant from them is a parabola.
The line
perpendicular to the directrix and passing through the focus (that is, the line
that splits the parabola through the middle) is called the "axis of
symmetry". The point on the axis of symmetry that intersects the parabola
is called the "vertex", and it is the point where the curvature is greatest.
Parabolas can open up, down, left, right, or in some other arbitrary direction.
Any parabola can be repositioned and rescaled to fit exactly on any other
parabola - that is, all parabolas are similar.
Equation
in Cartesian coordinates
Let the directrix be the line x = −p and let the focus be the
point (p, 0). If (x, y) is a point on the
parabola then, by Pappus' definition of a parabola, it is the same distance
from the directrix as the focus; in other words:
Squaring both sides and simplifying
produces
�
as
the equation of the parabola.
By translation, the above is the general
equation of a parabola with a horizontal axis; by interchanging the roles of x and y one
obtains the corresponding equation of a parabola with a vertical axis as
(Our solved example in mathguru.com uses
this concept)
Equations
Cartesian
Vertical axis of symmetry
(Our solved example in mathguru.com uses
this concept)
where
.
Parametric form:
http://en.wikipedia.org/wiki/Parabola
The above explanation is copied from
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