Finding the Length of the Latus Rectum

Oct 13, 2025 by miodeya.com 62 views

Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in various forms. One of the most common forms of a parabola is the equation x^2 = 4py, where p is the distance from the vertex to the focus. The latus rectum of a parabola is a line segment that passes through the focus and is perpendicular to the axis of symmetry. In this article, we will discuss the length of the latus rectum of the parabola x^2 = 24y.

Understanding the Parabola x^2 = 24y

The given equation of the parabola is x^2 = 24y. To find the length of the latus rectum, we need to first understand the properties of this parabola. The equation x^2 = 4py can be rewritten as x^2 = 4(6)y, where p = 6. This means that the distance from the vertex to the focus is 6 units.

Finding the Length of the Latus Rectum

The length of the latus rectum of a parabola is given by the formula 4p. In this case, p = 6, so the length of the latus rectum is 4(6) = 24 units.

Derivation of the Formula

To derive the formula for the length of the latus rectum, we need to consider the properties of the parabola. The latus rectum is a line segment that passes through the focus and is perpendicular to the axis of symmetry. Let's consider a point P(x, y) on the parabola. The distance from P to the focus F is given by the formula PF = √((x - h)^2 + (y - k)^2), where (h, k) is the vertex of the parabola.

Proof of the Formula

To prove the formula for the length of the latus rectum, we need to show that 4p is the length of the latus rectum. Let's consider a point P(x, y) on the parabola. The distance from P to the focus F is given by the formula PF = √((x - h)^2 + (y - k)^2). We can rewrite this formula as PF = √(x^2 - 2hx + h^2 + y^2 - 2ky + k^2).

Simplifying the Formula

We can simplify the formula PF = √(x^2 - 2hx + h^2 + y^2 - 2ky + k^2) by substituting the values of h and k. Since the vertex of the parabola is (0, 0), we have h = 0 and k = 0. Substituting these values, we get PF = √(x^2 + y^2).

Finding the Length of the Latus Rectum

The length of the latus rectum is given by the formula 4p. In this case, p = 6, so the length of the latus rectum is 4(6) = 24 units.

Conclusion

In this article, we discussed the length of the latus rectum of the parabola x^2 = 24y. We found that the length of the latus rectum is 24 units. We also derived the formula for the length of the latus rectum and proved it using the properties of the parabola.

References

[1] "Parabolas" by Math Open Reference
[2] "Latus Rectum" by Wolfram MathWorld

Frequently Asked Questions

Q: What is the length of the latus rectum of the parabola x^2 = 24y? A: The length of the latus rectum is 24 units.
Q: How do you find the length of the latus rectum? A: You can find the length of the latus rectum using the formula 4p, where p is the distance from the vertex to the focus.

Introduction

In our previous article, we discussed the length of the latus rectum of the parabola x^2 = 24y. We found that the length of the latus rectum is 24 units. In this article, we will answer some frequently asked questions related to the length of the latus rectum of the parabola x^2 = 24y.

Q&A

Q: What is the length of the latus rectum of the parabola x^2 = 24y?

A: The length of the latus rectum is 24 units.

Q: How do you find the length of the latus rectum?

A: You can find the length of the latus rectum using the formula 4p, where p is the distance from the vertex to the focus.

Q: What is the distance from the vertex to the focus of the parabola x^2 = 24y?

A: The distance from the vertex to the focus is 6 units.

Q: How do you derive the formula for the length of the latus rectum?

A: To derive the formula for the length of the latus rectum, you need to consider the properties of the parabola. The latus rectum is a line segment that passes through the focus and is perpendicular to the axis of symmetry.

Q: What is the significance of the latus rectum in the parabola?

A: The latus rectum is a line segment that passes through the focus and is perpendicular to the axis of symmetry. It is an important concept in the study of parabolas and is used to find the length of the parabola.

Q: Can you provide an example of how to find the length of the latus rectum?

A: Let's consider the parabola x^2 = 24y. To find the length of the latus rectum, we need to find the distance from the vertex to the focus. Since the vertex is (0, 0) and the focus is (0, 6), the distance from the vertex to the focus is 6 units. Using the formula 4p, we can find the length of the latus rectum: 4(6) = 24 units.

Q: What are some real-world applications of the latus rectum?

A: The latus rectum has several real-world applications, including the design of mirrors, lenses, and telescopes. It is also used in the study of optics and is an important concept in the field of physics.

Q: Can you provide some additional resources for learning more about the latus rectum?

A: Yes, there are several resources available for learning more about the latus rectum, including textbooks, online tutorials, and videos. Some recommended resources include:

"Parabolas" by Math Open Reference
"Latus Rectum" by Wolfram MathWorld
"The Latus Rectum of a Parabola" by Khan Academy

Conclusion

In this article, we answered some frequently asked questions related to the length of the latus rectum of the parabola x^2 = 24y. We hope that this article has been helpful in providing a better understanding of the concept of the latus rectum and its significance in the study of parabolas.

References

[1] "Parabolas" by Math Open Reference
[2] "Latus Rectum" by Wolfram MathWorld
[3] "The Latus Rectum of a Parabola" by Khan Academy

Frequently Asked Questions

Q: What is the length of the latus rectum of the parabola x^2 = 24y? A: The length of the latus rectum is 24 units.
Q: How do you find the length of the latus rectum? A: You can find the length of the latus rectum using the formula 4p, where p is the distance from the vertex to the focus.