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Example:Finding Equation of Parabola

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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 (xy) 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)





Vertical axis of symmetry

(Our solved example in mathguru.com uses this concept)




Parametric form:




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.