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

Simplext Parabola Example
  • 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

Parabola Equations

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