Equation of the Parabola | TutorVista.com

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In mathematics, the parabola (plural parabolae or parabolas, pronounced /pəˈræbələ/, from the Greek παραβολή) is a conic section, the intersection of a right circular conical surface and a plane 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.
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 general equation of a parabola with a horizontal axis is

and interchanging the roles of x and y gives the corresponding equation of a parabola with a vertical axis as

The last equation can be rewritten

so the graph of any function which is a polynomial of degree 2 in x is a parabola with a vertical axis.
More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation of the form

such that

where all of the coefficients are real, neither A nor B is zero and more than one solution exists, defining a pair of points (x, y) on the parabola. That the equation is irreducible means it does not factor as a product of two not necessarily distinct linear equations.

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• 05
• conic
• parabola