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Home > Articles > Latus Rectum of Parabola, Ellipse, Hyperbola with Definition, Formula, Properties & Example
Updated on 25th April, 2023 , 8 min read
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.
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).
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.
The following table shows the latus rectum and the ends of latus rectums for different standard equations of a parabola.
Equation of Parabola |
Focus |
Latus Rectum |
Endpoints of Latus Rectum |
y2=4ax |
(a, 0) |
x = a |
(a, 2a), (a, -2a) |
y2=−4ax |
(-a, 0) |
x = -a |
(-a, 2a), (-a, -2a) |
x2=4ay |
(0, a) |
y = a |
(2a, a), (-2a, a) |
x2=−4ay |
(0, -a) |
y = -a |
(2a, -a), (-2a, -a) |
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.
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.
The summary for the latus rectum of all the conic sections is given below:
Conic Section |
Length of the Latus Rectum |
Ends 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) |
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:
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:
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).
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.
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).
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.
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).
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).
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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By - Nikita Parmar 2024-09-06 10:59:22 , 6 min readThe latus rectum of a parabola is the line segment perpendicular to the axis of symmetry and passing through the focus.
The length of the latus rectum of a parabola is equal to four times the focal length of the parabola. Alternatively, you can also use the formula L = 2a, where a is the distance between the vertex and focus of the parabola.
The axis of symmetry of a parabola is the vertical line passing through its vertex, while the latus rectum is a line segment perpendicular to the axis of symmetry and passing through the focus.
A parabola has only one latus rectum, which is perpendicular to its axis of symmetry and passes through its focus.
The latus rectum is an important geometric property of a parabola because it determines the size and shape of the parabolic curve. It is also useful in many applications such as optics, where it describes the focal length of a parabolic mirror.
The length of the latus rectum of a parabola can be derived using the geometric definition of the parabola and the properties of its focus and directrix. By using the distance formula and solving for the length of the line segment that passes through the focus and is perpendicular to the axis of symmetry, we arrive at the formula L = 4f, where f is the focal length of the parabola.
The latus rectum is a line segment that is perpendicular to the axis of symmetry of a parabola and passes through its focus. It is also parallel to the directrix, which is a horizontal line located at a distance of f units below the vertex of the parabola, where f is the focal length.
No, the latus rectum of a parabola cannot be greater than the length of its axis because the axis is the longest dimension of the parabolic curve. The latus rectum is only a line segment that is perpendicular to the axis and passes through the focus.
The focus of a parabola can be found by using the geometric definition of the parabola and the properties of its latus rectum. By using the distance formula and solving for the distance between the vertex and focus, we arrive at the formula f = L/4, where L is the length of the latus rectum.
No, the length of the latus rectum of a parabola cannot be negative because it is defined as a non-negative distance between two points on the parabolic curve. Additionally, the latus rectum is a physical line segment and cannot have a negative length.