Latus Rectum of Parabola, Ellipse, Hyperbola with Definition, Formula, Properties & Example

Latus Rectum of Parabola Overview

The latus rectum of a parabola is a line segment that passes through the focus and is perpendicular to the axis of the parabola. It intersects the parabola at two distinct points and is also known as a focal chord. In the case of a parabola with equation y2 = 4ax, the length of the latus rectum is 4a units, and its endpoints are located at (a, 2a) and (a, -2a).

In this article, we will explore the latus rectum of a parabola in greater detail, discussing its properties, and related terms, and providing examples to aid in understanding. Additionally, we will answer frequently asked questions about this topic.

What is a Parabola?

A parabola is a curve that is defined by a set of points that are equidistant to a fixed point (known as the focus) and a fixed line (known as the directrix). The shape of a parabola is such that it is symmetrical around an axis known as the axis of symmetry. The vertex of the parabola is the point where the axis of symmetry intersects the parabola.

The standard equation for a parabola is given by y = ax² + bx + c, where a, b, and c are constants. The value of the coefficient determines whether the parabola opens upward (if a > 0) or downward (if a < 0).

What Is Latus Rectum of Parabola?

The latus rectum of a parabola is a line segment that passes through the focus of the parabola and is perpendicular to its axis. It can also be understood as the focal chord parallel to the directrix of the parabola. This property of the latus rectum is derived from the definition of a parabola as the set of all points in a plane that are equidistant to a fixed point, called the focus, and a fixed line called the directrix.

For a parabola in standard form, y2 = 4ax, the length of the latus rectum is equal to LL' = 4a. This length is derived using the distance formula between two points. The two endpoints of the latus rectum are located at (a, 2a) and (a, -2a), which are found by substituting x = a into the equation of the parabola and solving for y.

The latus rectum of a parabola has an interesting property where its endpoints and the focus of the parabola lie on the same straight line. Furthermore, the length of the latus rectum is equivalent to the distance between its two endpoints.

Standard Equation for Latus Rectum of Parabola

The following table shows the latus rectum and the ends of latus rectums for different standard equations of a parabola.

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Length of Latus Rectum of Parabola

Let the ends of the latus rectum of the parabola, y2=4ax be L and L'. The x-coordinates of L and L' are equal to ‘a' as S = (a, 0)

Assume that L = (a, b).

We know that L is a point of the parabola, we have

b2 = 4a (a) = 4a2

Take square root on both sides, we get b = ±2a

Therefore, the ends of the latus rectum of a parabola are L = (a, 2a), and L' = (a, -2a)

Hence, the length of the latus rectum of a parabola, LL' is 4a.

Length of Latus Rectum of Hyperbola

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

The ends of the latus rectum of a hyperbola are (ae, ±b2/a2), and the length of the latus rectum is 2b2/a.

Latus Rectum of Conic Sections

The summary for the latus rectum of all the conic sections is given below:

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Properties of the Latus Rectum

The latus rectum of a parabola is an important characteristic of the shape and geometry of a parabolic curve. Here are some key properties of the latus rectum of a parabola:

1. Collinearity:The endpoints of the latus rectum and the focus of the parabola are all collinear. This means that they lie in the same straight line. This is an important characteristic of the latus rectum, and it can be used to determine the position of the focus and the endpoints of the latus rectum.
2. Length: The length of the latus rectum of a parabola is equal to 4 times the distance between the focus and the directrix of the parabola. It is also equal to the distance between the two endpoints of the latus rectum. In an equation of the form y² = 4ax, where a is the distance between the vertex and the focus of the parabola, the length of the latus rectum is 4a.
3. Perpendicularity:The latus rectum is perpendicular to the axis of the parabola. This means that the latus rectum intersects the axis of the parabola at a right angle.
4. Symmetry: The latus rectum divides the parabola into two equal parts. The part of the parabola on one side of the latus rectum is a mirror image of the part of the parabola on the other side of the latus rectum.
5. Focal property:The latus rectum is also known as the focal chord. The chord of the parabola passes through the parabola's focus and is parallel to the directrix of the parabola.
6. Focus and vertex: The midpoint of the latus rectum is the vertex of the parabola. The focus of the parabola is located at a distance of a unit from the vertex, where a is the distance between the vertex and the focus.

Terms Related to Latus Rectum of Parabola

There are several terms related to the latus rectum of a parabola that are important to understand. Here are some of the most important terms related to the latus rectum:

1. Focus of Parabola: The focus of a parabola is a fixed point on the parabolic curve. The latus rectum of a parabola passes through the focus of the parabola and is parallel to the directrix of the parabola.
2. Directrix of Parabola: The directrix of a parabola is a fixed line that is equidistant from the focus of the parabola and perpendicular to the axis of the parabola. The latus rectum of a parabola is parallel to the directrix of the parabola.
3. Axis of Parabola: The axis of a parabola is the line that passes through the vertex of the parabola and is perpendicular to the directrix of the parabola. The latus rectum of a parabola is perpendicular to the axis of the parabola.
4. Vertex of Parabola: The vertex of a parabola is the point where the axis of the parabola intersects the parabolic curve. The midpoint of a parabola's latus rectum is the parabola's vertex.
5. Length of Parabola: The length of the latus rectum of a parabola is equal to 4 times the distance between the focus and the directrix of the parabola. In an equation of the form y² = 4ax, where a is the distance between the vertex and the focus of the parabola, the length of the latus rectum is 4a.
6. Midpoint of Parabola: The midpoint of the latus rectum of a parabola is the point where the latus rectum intersects the axis of the parabola. This point is also the vertex of the parabola.
7. Endpoint of Parabola: The endpoints of the latus rectum of a parabola are the two points where the latus rectum intersects the parabolic curve. For a parabola y² = 4ax, the endpoints of the latus rectum are (a, 2a) and (a, -2a).

Examples of latus rectum of parabola

Find the latus rectum of the parabola y^2 = 16x.

Solution: Here, we have a = 4. Therefore, the length of the latus rectum is 4a = 16 units. The endpoints of the latus rectum are (a, 2a) and (a, -2a). Substituting the value of a, we get the endpoints as (4, 8) and (4, -8). Thus, the latus rectum is a horizontal line passing through the focus, which is (4, 0).

Find the equation of the latus rectum of the parabola y^2 = 8x.

Solution:Here, we have a = 2. Therefore, the length of the latus rectum is 4a = 8 units. The endpoints of the latus rectum are (a, 2a) and (a, -2a). Substituting the value of a, we get the endpoints as (2, 4) and (2, -4). Thus, the latus rectum is a horizontal line passing through the focus, which is (2, 0). Hence, the equation of the latus rectum is x = 2.

Find the latus rectum of the parabola x^2 = -4y.

Solution:Here, we have a = -1. Therefore, the length of the latus rectum is 4a = -4 units. The endpoints of the latus rectum are (a, 2a) and (a, -2a). Substituting the value of a, we get the endpoints as (-1, -2) and (-1, 2). Thus, the latus rectum is a vertical line passing through the focus, which is (0, -1).

Find the equation of the latus rectum of the parabola x^2 = 16y.

Solution:Here, we have a = 4. Therefore, the length of the latus rectum is 4a = 16 units. The endpoints of the latus rectum are (a, 2a) and (a, -2a). Substituting the value of a, we get the endpoints as (4, 8) and (4, -8). Thus, the latus rectum is a vertical line passing through the focus (0, 4). Hence, the equation of the latus rectum is y = 4.

Find the latus rectum of the parabola y^2 = -12x.

Solution:Here, we have a = -3. Therefore, the length of the latus rectum is 4a = -12 units. The endpoints of the latus rectum are (a, 2a) and (a, -2a). Substituting the value of a, we get the endpoints as (-3, -6) and (-3, 6). Thus, the latus rectum is a horizontal line passing through the focus (3, 0).

Find the equation of the latus rectum of the parabola y^2 = -24x.

Solution: Here, we have a = -6. Therefore, the length of the latus rectum is 4a = -24 units. The endpoints of the latus rectum are (a, 2a) and (a, -2a). Substituting the value of a, we get the endpoints as (-6, -12) and (-6, 12).

Find the length of the latus rectum of an ellipse 4x2 + 9y2 – 24x + 36y – 72 = 0.

Solution: Given: 4x2 + 9y2 – 24x + 36y – 72 = 0

⇒(4x2 – 24x) + (9y2 + 36y) – 72 = 0

⇒4(x2 -6x) + 9(y2 + 4y) – 72 = 0

⇒4[x2 – 6x +9] + 9[y2 + 4y +4] = 144

⇒4(x – 3)2 + 9(y + 2)2 = 144

⇒{(x – 3)2/ 36} + {(y + 2)2/ 16} = 1

⇒{(x – 3)2/ 62} + {(y + 2)2/ 42} = 1

⇒a = 3 and b = 2

Therefore, the length of the latus rectum of an ellipse is given as:

= 2b2/a

= 2(2)2 /3

= 2(4)/3

= 8/3

Hence, the length of the latus rectum of the ellipse is 8/3.

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