Length of Latus Rectum of Parabola Formula

Here you will learn formula to find the length of latus rectum of parabola with examples.

Let’s begin –

Latus Rectum of Parabola

A double ordinate through the focus is called the latus rectum i.e. the latus rectum of a parabola is a chord passing through the focus perpendicular to the axis.

In the given figure, LSL’ is the latus rectum of the parabola $y^2$ = 4ax.

By the symmetry of the curve SL = SL’ = $\lambda$ (say). So, the coordinates of L are $(a, \lambda)$.

Since L lies on $y^2$ = 4ax, therefore

${\lambda}^2$ = $4a^2$  $\implies$  $\lambda$ = 2a

$\implies$  LL’ = $2\lambda$ = 4a

Hence, Latus Rectum = 4a

Note : The length of latus rectum of all other forms of parabola i.e. $x^2$ = 4ay , $y^2$ = -4ax and $x^2$ = -4ay is also equal to 4a.

Also Read : Different Types of Parabola Equations

Coordinates of Latus Rectum

The coordinates of L and L’ , end points of the latus rectum, are (a, 2a) and (a, -2a) respectively.

Example : For the given parabola, find the length of the latus rectum:

(i) $y^2$ = 8x

(ii) $x^2$ = -16y

Solution :

(i) The given parabola is of the form $y^2$ = 4ax, where 4a = 8 i.e. a = 2.

Hence, Length of latus rectum = 4a = 8

(ii) The given parabola is of the form $x^2$ = -4ay, where 4a = 16 i.e. a = 4.

Hence, Length of latus rectum = 4a = 16

Example : Find the latus rectum of the parabola $y^2 – 8y – x + 19$ = 0

Solution : The given equation is

$y^2 – 8y – x + 19$ = 0  $\implies$  $y^2 – 8y$ = x – 19

$\implies$  $y^2 – 8y + 16$ = x – 19 + 16

$\implies$  $(y – 4)^2$ = (x – 3)

The equation is of the form $y^2$ = 4ax, where 4a = 1 i.e. a = 1/4.

Hence, Length of Latus Rectum is 4a = 1.