How To Find The Equation Of The Parabola When Latus Rectum Is Given

A parabola is a plane curve formed by a point moving such that its distance from a fixed point is equal to its distance from a fixed-line. The fixed-line is the directrix of the parabola and the fixed point F is the focus. As far as JEE is concerned, parabola is an important topic in conics. In this article, we will learn how to find the equation of the parabola when latus rectum is given.

What is Latus Rectum of Parabola?

Latus Rectum is a line segment perpendicular to the axis of the parabola, through the focus and whose endpoints lie on the parabola.

The length of the latus rectum is given by 4a.

Standard Equation

The equation of the parabola with vertex at the origin, focus at (a,0) and directrix x = -a is

y2 = 4ax.

Solved Examples

Example 1

Find the equation of the parabola with latus rectum joining points (4, 6) and (4,-2).

Solution:

Given latus rectum joining the points (4, 6) and (4, -2).

So the length of latus rectum = √[(4-4)2 + (-2-6)2]

= √64

= 8

So 4a = 8

Equation of parabola is y2 = 4ax.

y2 = 8x is the required equation.

Example 2.

Find the equation of the parabola whose focus is at (3,0) and the length of the latus rectum is 12.

Solution:

Given length of latus rectum, 4a = 12

So a = 12/4 = 3

Equation of parabola is y2 = 4ax

y2 = 12x is the required equation.

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