# Parabola: Standard Equation of Parabola and Parabola Equation Derivation (For CBSE, ICSE, IAS, NET, NRA 2022)

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# Title: Parabola

• A parabola is a U-shaped plane curve where any point is at an equal distance from a fixed point (known as the focus) and from a fixed straight line which is known as the Directrix
• A parabola, plural “parabolas;” Gray 1997, S the set of all points in the plane equidistant from a given line L (the conic section directrix) and a given point F not on the line (the focus) .
• Focal parameter (i.e.. , the distance between the directrix and focus) is therefore given by 2-a, where a is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid.
• Parabola is an integral part of conic section topic and its entire concepts parabola are covered here.

## What is Parabola?

• Section of a right circular cone by a plane parallel to a generator of the cone is a parabola.
• It is a locus of a point, which moves so that distance from a fixed point (focus) is equal to the distance from a fixed line (Directrix)
• Fixed point is called focus
• Fixed line is called Directrix

## Standard Equation of Parabola

• The simplest equation of a parabola is when the directrix is parallel to the y-axis.
• Generally, if the directrix is parallel to the y-axis in the standard equation of a parabola is given as:
• If the parabola is sideways i.e.. , the directrix is parallel to x-axis, the standard equation of a parabola becomes,
• Apart from these two, the equation of a parabola can also be and if the parabola is in the negative quadrants.
• Thus, the four equations of a parabola are given as:

## Parabola Equation Derivation

• Three points uniquely determine one parabola with directrix parallel to the x-axis and one with directrix parallel to the y-axis.
• If these parabolas pass through the three points , they are given by equations

and

## Parabola Equations

In polar coordinates, the equation of a parabola with parameter and center (0,0) is given by