Latus Rectum of Conic Section

In Mathematics, a conic section is represented as a curve which we get from the intersection of the surface of a cone. There are varied types of conic sections. These are parabola, ellipse, and hyperbola. To represent these curves, many important terms are used such as focus, directrix, latus rectum, locus, asymptote, etc. In this article, we will study about latus rectum, latus rectum definitions, latus rectum examples, and latus rectum of conic sections.


Latus Rectum of Conic Section Introduction

The latus rectum of conic section is stated as the chord that passes through the focus and is perpendicular to the major axis and includes both endpoints on the curve.

The length of the latus rectum is specified differently for each conic section:

  • Latus rectum is always equivalent to the length of the diameter in a circle.

  • The length of the latus rectum in a parabola is equal to the four times the focal length.

  • The length of the latus rectum in hyperbola is equal to twice the square of the length of the transverse axis divided by the length of the conjugate axis.

(image will be uploaded soon)


Latus Rectum Definition

In the conic section, the latus rectum is the chord drawn through the focus and parallel to the directrix. The word latus is derived from the Latin word “ latus’’ which implies side and the term “rectum” meaning straight. Half portion of the latus rectum is known as the semi latus rectum. The diagram given below represents the latus rectum of a parabola.(image will be uploaded soon)


Length of the Latus Rectum of a Parabola

Let us consider the length of the latus rectum of the parabola y2= 4ax as L and L’. The x coordinates of L and L’ are equivalent to “a” as S = (a, 0)

Let us assume L = (a, b)

As we know, L is the point of parabola. Accordingly, we have

= 4a (a) = 4a²

Taking square root on both LHS and RHS, we get b equivalent to ± 2a

Hence, the end of the latus rectum of a parabola are L = (a, 2a) and L’ (a, -2a)

Thus, the length of the latus rectum of a parabola, L L’’ is 4a.


Length of the Latus Rectum of Hyperbola

Latus rectum of a hyperbola is defined symmetrically as in the case of ellipse and parabola.


The End of the Latus Rectum of a Parabola are Termed as (ae, ±b2 / a2) and the Length of the Latus Rectum is Termed as 2b2/a

Conic Section 

Length of the Latus Rectum

End of the Latus Rectum

y²= 4ax

4a

L= ( a, 2a) , L’ (a, -2a)

(x² / a²) + (y² /b²)= 1

If a > b, 2b² / a

L = (ae, b² a) L’ (ae- b² a)

(x² / a²) + (y² /b²)= 1

If b > a, 2a² / b

L = (ae, b² a) L’ (ae- b² a)

(x² / a²) - (y² /b²)= 1

2b²/ a

L = (ae, b² a) L’ (ae- b² a)


Latus Rectum Examples

1. What will be the length of the latus rectum whose parabola equation is y2 = 12x

Solution:

y2 = 12x

y2 = 4 (3) x

As y2 = 4ax is the equation of a parabola, we get the value of a

Therefore, the value of a=3

Thus, the length of the latus rectum of a parabola is 4a= 4(3) = 12.


Solved Example

1. What will be the length of the latus rectum of the following parabola x2 = - 4y.


Solution: From the equation given above, we can conclude that parabola is symmetric about the Y axis and it is open in a downward position.

x2 = - 4y

x2 = - 4ay

4a = 4

Thus, the length of the latus rectum of a given parabola is 4 units.


2. Calculate the length of the latus rectum of the following parabola x2- 2x + 8y + 17= 0.


To calculate the value a, we will first convert the above equation in standard form.

x2 - 2x = - 8y -17= 0

x2 - 2x(1) + 12 -12= - 8y -17

(x-1)2 -  1 = - 8y -17 + 1

(x-1)2 = -8y-16

(x-1)2 = 8(y +2)

From this equation, we can conclude that the given parabola is symmetric about y axis and it is open in an upward position.

Length of the latus rectum = 4a

4a=8

Thus, the length of the latus rectum of the given parabola is 8 units.


Facts

  • Conic section was introduced by Menaechmus around 360 - 350 B.C

  • Euclid gave the name to the parabola around 300 BC.


Quiz Time

1. The length of the latus rectum of hyperbola \[\frac{x}{16}\] - \[\frac{y}{19}\] equals to 1 is

  1. 32/3

  2. 9/2

  3. 8/3

  4. None of these


2. The length of the latus rectum of hyperbola XY = c² is

  1. 2 c

  2. 4 c

  3. 2\[\sqrt{2}\]c

  4. \[\sqrt{2}\]c

FAQ (Frequently Asked Questions)

1. Explain the Conic Section.

Circle, Hyperbola, parabola, ellipse are different types of curves in Geometry. These curves are determined as conic sections or conics as we get these curves form the intersection of a plane with a double-napped right circular cone. These curves are used in various fields such as automobiles, headlights, antennas, telescopes, and reflectors.

We get the following situation when the plane cuts the cone other than the vertex.

  • When β= 90°, the section is determined as a circle

  • When α < β < 90°, the section is determined as an ellipse

  • When α = β, the section is determined as a parabola

  • When 0 ≤ β < α, the section is determined as a hyperbola

  • Where β is considered as the angle formed by the plane through the vertical axis of the cone.

2. Explain the Latus Rectum of an Ellipse.

In terms of locus, an ellipse is the set of all those points on XY- plane, whose distance from the two fixed points ( also known as foci) sum up to a constant value.

The line segment perpendicular to the major axis passing through any of the foci such that their endpoint lies on the ellipse is known as the latus rectum of an ellipse.

The length of the latus rectum of an ellipse is 2b²/a

Length = 2b²/a

 Here a and b are considered as the length of the minor and major axis.