A parabola is locus of a point which moves in a plane, such that its distance from a fixed point called focus is equal to its perpendicular distance from a fixed straight line called directrix. Graph of a Parabola and their types are shown below.

## Basic Concepts of a Parabola

**(a) Focal distance :** The distance of a point on parabola from the focus is called focal distance of the point.

**(b) Focal chord : **A chord of parabola, which passes through focus is called focal chord.

**(c) Double ordinate : **A chord of the parabola perpendicular to the axis of the symmetry is called double ordinate.

**(d) Latus rectum : **A double ordinate passing through focus or a focal chord perpendicular to axis of parabola is called latus rectum.

## Types and Graph of a Parabola :

Four standard forms of parabola are \(y^2\) = 4ax ; \(y^2\) = -4ax ; \(x^2\) = 4ay ; \(x^2\) = -4ay. All of them with their graphs are given below :

**(i) Parabola \(y^2 = 4ax\)** :

Vertex O is (0, 0)

Focus S is (a, 0)

Axis is y = 0

Directrix ZZ’ is x = -a

Focal length = x + a

Length of Latus rectum = 4a

(ii) ** Parabola \(y^2 = -4ax\)** :

Vertex O is (0,0)

Focus S is (-a,0)

Axis is y = 0

Directrix ZZ’ is x = a

Focal length = x – a

Length of Latus rectum = 4a

(iii) **Parabola \(x^2 = 4ay\)** :

Vertex O is (0,0)

Focus S is (0,a)

Axis is x = 0

Directrix ZZ’ is y = -a

Focal length = y + a

Length of Latus rectum = 4a

(iv) **Parabola \(x^2 = -4ay\)** :

Vertex O is (0,0)

Focus S is (0,-a)

Axis is x = 0

Directrix ZZ’ is y = a

Focal length = y – a

Length of Latus rectum = 4a

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