# Latus Rectum

In a conic section, the latus rectum is the chord (or line segment) passing through the focus. It is perpendicular to the major axis and parallel to the directrix,  having both endpoints on the curve. The word ‘latus rectum’ came from the Latin word ‘latus’ meaning ‘side’, and ‘rectum’ meaning ‘straight’. The half of the latus rectum is the semi latus rectum.

We will learn how to find the latus rectum of the parabola, ellipse, and hyperbola.

## Formulas

### Latus Rectum of Parabola

The latus rectum in a parabola is the chord passing through its focus and perpendicular to its axis. It is also the focal chord parallel to the directrix. A parabola has only one latus rectum.

The formula is given below:

For the standard equation of a parabola, y2 = 4ax,

Length of latus rectum = 4a,

Endpoints of latus rectum = (a, 2a), and (a, -2a)

### Latus Rectum of Ellipse

The latus rectum in an ellipse is the chord passing through its foci and perpendicular to its major axis. It is also the focal chord parallel to the directrix. An ellipse has two latus rectums.

The formula is given below:

For the standard equation of an ellipse, ${ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1}$,

Length of latus rectum for a > b = 2b2/a

Length of latus rectum for b > a = 2a2/b

For foci = (ae, 0), endpoints of the latus rectum  = (ae, b2/a), here e = eccentricity

For foci = (-ae, 0), endpoints of the latus rectum = (-ae, -b2/a).

The value of eccentricity ‘e’ of the ellipse lies between 0 and 1, (0 < e < 1).

${e=\sqrt{1-\dfrac{b^{2}}{a^{2}}}}$

The endpoints of latus rectum and the focus are collinear.

### Latus Rectum of Hyperbola

The latus rectum in a hyperbola is the chord passing through its foci and perpendicular to its major axis. It is also the focal chord parallel to the directrix. A hyperbola has two latus rectums.

The formula is given below:

For the standard equation of an hyperbola,  ${\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1}$

Length of latus rectum = 2b2/a

For foci = (ae, 0),  endpoints of the latus rectum  = (ae, b2/a), here e = eccentricity

For foci = (-ae, 0), endpoints of the latus rectum = (-ae, -b2/a).

The value of eccentricity ‘e’ of the hyperbola is always greater than 1, (e  >  1)

${e=\sqrt{1+\dfrac{b^{2}}{a^{2}}}}$

The endpoints of latus rectum and the focus are collinear.

## Semi-Latus Rectum

Semi-latus rectum is the chord passing through the focus of a conic section parallel to the directrix and perpendicular to the axis, and its endpoint is on the curve. It is half of the latus rectum.

## Solved Examples

Find the length of the latus rectum of the parabola y2 = 12x. Also find its endpoints.

Solution:

As we know,
For the standard equation of a parabola, y2 = 4ax,
Here, the standard equation of the parabola isÂ  y2 = 12x
Length of latus rectum = 4a,
âˆ´4a = 12 units
Now,
We know, 4a = 12
âˆ´a = 3
Endpoints of latus rectum = (a, 2a), and (a, -2a), here a = 3
âˆ´Endpoints of latus rectum = (3, 6) and (3, -6)

Find the length of the latus rectum of the ellipse x2/16 + y2/49 = 1. Also find its endpoints.

Solution:

Here, a2 = 16
âˆ´a = 4
b2 = 49
âˆ´b = 7
So, bÂ  >Â  a
As we know,
For the standard equation of an ellipse,
Length of latus rectum for bÂ  >Â  a = 2a2/b
= 2 Ã— 16/7
= 32/7 units
e = âˆš(1 – b2/a2)
=Â  âˆš(1 – 49/16)
= -2.06
ae = 4 Ã— (-2.06)
= -8.24
As we know,
Endpoints of the latus rectum of ellipseÂ  = (ae, b2/a), and (-ae, -b2/a)
Now, b2/a = 49/4 = 12.2
= (-8.24, 12.2) and (8.24, 12.2)

## One thought on “Latus Rectum”

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