**Definition of Parabola**

If *S* and *L* are a point and a fixed straight line (not passing through *S*) in a plane, respectively, and a point *P* in that plane is moving in such a way that the point *P* is always equidistant from the point *S* and the straight line *L* in all locations, then the locus of point *P* is called Parabola.

The fixed point *S* is called the **focus** of the parabola and the fixed straight line *L* is called the **directrix** of the parabola.

The perpendicular from focus *S* to the directrix *L* is called the **axis** of the parabola.

The point where the parabola intersects with its axis is called the **vertex**.

Any chord of the parabola passing through the focus is called the **focal chord**.

The length of all the chords perpendicular to the axis of the parabola is called **double ordinate**.

The double ordinate passing through the focus of the parabola is called the **latus rectum**.

In the above figure straight line *L* – directrix, point *S* – focus, straight-line *AS* – axis, point *A* – vertex, *PN* = ordinate of point *P*, *PNQ* = double ordinate of the point *P*, *AN* = abscissa of the point *P*, *L _{1}L_{2}* = latus rectum,

*PSR*= chord passing through focus.

The standard equation of a parabola is {{y}^{2}}=4ax where ‘a’ is the between focus and the vertex of the parabola and x-axis is the axis of the parabola.

**Some Formulae on Parabola**

1. In the following a = is the between focus and the vertex of the parabola

Equation of Parabola | Coordinate of the Vertex | Axis | Coordinate of the Focus | Length of the Latus Rectum | Equation of the Directrix |

\[{{y}^{2}}=4ax\] | (0, 0) | Positive x-axis | (a, 0) | 4a units | x + a = 0 |

\[{{y}^{2}}=-4ax\] | (0, 0) | Negative x-axis | (–a, 0) | 4a units | x – a = 0 |

\[{{x}^{2}}=4ay\] | (0, 0) | Positive y-axis | (0, a) | 4a units | y + a = 0 |

\[{{x}^{2}}=-4ay\] | (0, 0) | Negative y-axis | (0, –a) | 4a units | y – a = 0 |

\[{{\left( y-\beta \right)}^{2}}=4a\left( x-\alpha \right)\] | (α, β) | Parallel to x-axis | (a + α, β) | 4a units | x + a = α |

\[{{\left( x-\alpha \right)}^{2}}=4a\left( y-\beta \right)\] | (α, β) | Parallel to y-axis | (α, a + β) | 4a units | y + a = β |

2. x=a{{y}^{2}}+by+c\,\,\left( a\ne 0 \right) represents the equation of a parabola with its axis parallel to x-axis.

3. y=p{{x}^{2}}+qx+r\,\,\left( p\ne 0 \right) represents the equation of a parabola with its axis parallel to y-axis.

4. P(x_{1}, y_{1}) point will be inside, outside or on the parabola {{y}^{2}}=4ax if the value of \left( y_{1}^{2}=4a{{x}_{1}} \right) > or < or = 0 respectively.

5. We can take any point P on {{y}^{2}}=4ax as \left( a{{t}^{2}},2at \right).

6. The parametric equation of {{y}^{2}}=4ax is x=a{{t}^{2}},y=2at. Here t is the parameter.

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