Latus Rectum

When studying conic sections, one important geometric property that comes up in the context of ellipses, hyperbolas, and parabolas is the Latus Rectum. The term "Latus Rectum" refers to a line segment that is drawn perpendicular to the major axis of a conic section, passing through a focus. It has distinct properties depending on the type of conic—ellipse, hyperbola, or parabola.

In this blog, we'll explore the Latus Rectum of Ellipse, Latus Rectum of Hyperbola, Latus Rectum of Parabola, and the related Semi Latus Rectum to understand their roles in conic sections.

1.0What is Latus Rectum?

In simple terms, the Latus Rectum is a line segment that passes through a focus of a conic section and is perpendicular to the major axis. The length of this line segment varies for different conic sections, and it plays a crucial role in their geometric properties.

2.0Latus Rectum Of Parabola

A parabola is a simpler conic section compared to the ellipse and hyperbola. It is defined as the set of points equidistant from a fixed point called the focus and a fixed line called the directrix. The Latus Rectum of Parabola refers to the line segment perpendicular to the axis of symmetry that passes through the focus of the parabola.

For a parabola with the standard equation $y^2=4ax$, the length of the Latus Rectum of Parabola is given by:

Latus of a rectum of Parabola = 4a

This formula helps to understand the width of the parabola at its focus, providing an important reference for its geometric properties.

3.0Latus Rectum Of Ellipse

An ellipse is defined as the set of all points such that the sum of the distances from 2 fixed points (called foci) to any point on the ellipse is constant. The Latus Rectum of Ellipse is the chord that goes through a focus and is perpendicular to the major axis. The length of the Latus Rectum of an ellipse depends on the semi-major axis (a) and the semi-minor axis (b).

For an ellipse with foci at (±c, 0) where $c^2=a^2-b^2$, the length of the Latus Rectum of Ellipse is given by the formula:

$\text{Latus Rectum Of Ellipse}=\frac{2b^2}{a}$

This formula tells us how wide the ellipse is at the focus. It helps in understanding the shape of the ellipse in relation to the distance between the foci and the lengths of the axes.

4.0Latus Rectum Of Hyperbola

A hyperbola is another type of conic section created by the intersection of a plane and a double cone. The Latus Rectum of Hyperbola behaves similarly to that of the ellipse, passing through a focus and being perpendicular to the transverse axis. However, because a hyperbola has two branches, the concept of the Latus Rectum is applied to both foci.

For a hyperbola with equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the Latus Rectum of Hyperbola is given by the formula:

$\text{Latus Rectum Of Hyperbola}=\frac{2b^2}{a}$

This shows that the length of the Latus Rectum in a hyperbola is also proportional to the axes but differs in how the branches of the hyperbola stretch.

5.0Semi Latus Rectum

The Semi Latus Rectum is simply half the length of the full Latus Rectum. In certain situations, particularly in orbital mechanics or when dealing with the focus-directrix property of conic sections, the Semi Latus Rectum is used to describe the distance from the focus to the curve, measured along the line of the Latus Rectum.

For any conic section, the Semi Latus Rectum (denoted as l) is given by:

• Parabola: $l=2a$
• Ellipse: $l=\frac{b^2}{a}$
• Hyperbola: $l=\frac{b^2}{a}$

The Semi Latus Rectum is a vital concept in physics, especially in celestial mechanics, where it often appears in the context of orbital equations.

6.0Sample Question on Latus Rectum

1. What is the Semi Latus Rectum?

Ans: The Semi Latus Rectum is half the length of the Latus Rectum. It is often used in physics and astronomy, especially in the study of orbits. The semi Latus Rectum (denoted by ll) is used in the following formulas:

• Parabola: $l=2a$
• Ellipse: $l=\frac{b^2}{a}$
• Hyperbola: $l=\frac{b^2}{a}$

The Semi Latus Rectum provides a simplified way to understand the properties of conic sections, especially in orbital mechanics.