# Parabola Formula

## Introduction

A conic is the locus of a point whose distance from a fixed point bears a constant ratio to its distance from a fixed line. The fixed point is the focus S and the fixed line is the directrix l

The constant ratio is called the eccentricity denoted by ‘e’

1. if 0 < e < 1, the conic is an ellipse.
2. e = 1, conic is a parabola
3. e > 1, conic is a hyperbola.

## Important Terms Related to Parabola

1. Axis:  A line perpendicular to thee directrix and passes through the focus.
2. Vertex: The intersection point of the conic and axis.
3. Centre: The point which bisects every chord of the conic passing through it.
4. Focal Chord: Any chord passing through the focus.
5. Latus Rectum: A double ordinate passing through the focus of the parabola.

## Standard Form of a Parabola and Related Terms

 Equation of Parabola y2 = 4ax Vertex (0, 0) Focus (a, 0) Equation of axis y = 0 Equation of directrix x + a = 0 Eccentricity e = 1 Extremities of Latus Rectum (a, 2a) and (a, -2a) Length of Latus Rectum 4a Parametric equation x = at2 and y = 2at Equation of Latus rectum x – a = 0

 Equation of Parabola y2 = – 4ax Vertex (0, 0) Focus (- a, 0) Equation of axis y = 0 Equation of directrix x – a = 0 Eccentricity e = 1 Extremities of Latus Rectum (-a, 2a) and (-a, -2a) Length of Latus Rectum 4a Parametric equation x = – at2 and y = 2at Equation of Latus rectum x + a = 0

 Equation of Parabola x2 = 4ay Vertex (0, 0) Focus (0, a) Equation of axis x = 0 Equation of directrix y + a = 0 Eccentricity e = 1 Extremities of Latus Rectum (2a, a) and (-2a, a) Length of Latus Rectum 4a Parametric equation x = 2at and y = at2 Equation of Latus rectum y – a = 0

 Equation of Parabola x2 = 4ay Vertex (0, 0) Focus (0, – a) Equation of axis x = 0 Equation of directrix y – a = 0 Eccentricity e = 1 Extremities of Latus Rectum (2a, – a) and (-2a, – a) Length of Latus Rectum 4a Parametric equation x = 2at and y = – at2 Equation of Latus rectum y + a = 0
 Equation of Parabola (y – β)2 = 4a (x – α) Vertex (α, β) Focus (a + α,  β) Equation of axis Parallel to x – axis Equation of directrix x + a = α Eccentricity 1 Extremities of Latus Rectum (α + a, β + 2a) and (α + a, β – 2a) Length of Latus Rectum 4a
 Equation of Parabola (x – α)2 = 4a (y – β) Vertex (α, β) Focus (α, a + β) Equation of axis Parallel to y – axis Equation of directrix y + a = β Eccentricity 1 Extremities of Latus Rectum (α + 2a, β + a) and (α – 2a, β + a) Length of Latus Rectum 4a

## Position of a Point

The point (x1, y1) lies outside, on or inside the parabola y2 = 4ax according as y1 – 4ax1 >, =, < 0.

 y1 – 4ax1 > 0 outside the parabola y1 – 4ax1 = 0 on the parabola y1 – 4ax1 < 0 inside the parabola
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