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## 7.5 Parabola

7.5.1 Introduction: A parabola is the locus of all points in a plane equidistant from a fixed point, called the focus, and a fixed straight line, called the directrix.

In the parabola shown in figure 2-8, point V, which lies halfway between the focus and the directrix, is called the vertex of the parabola. In this figure and in many of the parabolas discussed in the first portion of this section, the vertex of the parabola falls at the origin; however, the vertex of the parabola, like the center of the circle, can fall at any point in the plane. 7.5.2 Standard equation of Parabola:

### Take a point F(a, 0) as the focus, where a> 0.

A line D with equation x = -a is taken as directrix.

Let P(x, y) be a point on locus and PM is the perpendicular on the directrix.

Then,

PF = PM

PF2 = PM2

(x – a )2 + y2 = (x + a)2

y2 = (x + a)2 – (x – a)2

y2 = 4ax

this is the standard equation of parabola.

Focus = F(a, 0)

Vertex = O(0, 0)

Equation of axis; y = 0

Equation of directrix; x + a = 0

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