Finding The Focus Of A Parabola Given Vertex At Origin And Directrix Y=3

Jul 10, 2025 by Admin 73 views

Understanding Parabolas with Vertex at the Origin and Directrix y=3

In the realm of conic sections, parabolas hold a special place due to their unique properties and wide range of applications. Understanding the relationship between the vertex, focus, and directrix is crucial for mastering the characteristics of a parabola. This article delves into the specifics of a parabola with its vertex at the origin and a directrix defined by the equation y = 3. We will explore the fundamental principles governing parabolas and employ them to determine the coordinates of the focus.

Defining the Parabola: Vertex, Focus, and Directrix

At its core, a parabola is defined as the locus of points that are equidistant from a fixed point, known as the focus, and a fixed line, known as the directrix. The vertex of the parabola is the point where the parabola changes direction, and it lies exactly halfway between the focus and the directrix. This fundamental definition provides the cornerstone for understanding and analyzing parabolas.

When we consider a parabola with its vertex at the origin (0, 0), we establish a simplified coordinate system that allows us to focus on the core geometric relationships. The directrix, in this case, is given by the equation y = 3, which represents a horizontal line located 3 units above the x-axis. Given that the vertex is equidistant from the focus and directrix, and the vertex is at the origin, we can deduce that the focus must lie along the y-axis. Since the directrix is above the vertex, the focus must lie below the vertex to maintain the equal distance property. This initial deduction narrows down our search for the focus.

The distance between the vertex and the directrix is a critical parameter in defining the parabola's shape and orientation. In this scenario, the distance between the vertex (0, 0) and the directrix y = 3 is 3 units. Consequently, the distance between the vertex and the focus must also be 3 units. Since the focus lies below the vertex along the y-axis, its y-coordinate must be -3. Therefore, the coordinates of the focus are (0, -3). This careful application of the definition of a parabola allows us to precisely pinpoint the focus based on the given information about the vertex and directrix.

Determining the Focus of the Parabola

To effectively determine the focus of the parabola, let's revisit the fundamental properties that define this conic section. The parabola, as mentioned earlier, is the set of all points equidistant from the focus and the directrix. The vertex is the point where the parabola changes direction, and it sits precisely midway between the focus and the directrix. In this specific problem, we're given that the vertex is at the origin (0, 0) and the directrix is the horizontal line y = 3. This information is pivotal in locating the focus.

Since the vertex is at the origin and the directrix is a horizontal line y = 3, we can infer that the parabola opens downwards. The focus must lie on the axis of symmetry, which in this case is the y-axis because the directrix is a horizontal line. The distance between the vertex and the directrix is a crucial piece of information. The directrix y = 3 is 3 units above the vertex (0, 0). Therefore, the focus must be 3 units below the vertex to satisfy the condition that the vertex is equidistant from the focus and the directrix.

Given that the vertex is at (0, 0) and the focus is 3 units below it along the y-axis, the coordinates of the focus are (0, -3). This can be visualized easily on a coordinate plane. The directrix is a horizontal line at y = 3, the vertex is at the origin, and the focus is at (0, -3). Any point on the parabola will be equidistant from the point (0, -3) and the line y = 3. This understanding of the geometry of parabolas allows us to confidently determine the focus based on the given vertex and directrix.

The Correct Answer and Explanation

Based on the analysis, the correct answer is C. (0, -3). This is because the focus must lie on the axis of symmetry, which is the y-axis in this case, and it must be the same distance from the vertex as the directrix but in the opposite direction. Since the directrix is y = 3, the focus is at y = -3.

The other options can be eliminated as follows:

A. (0, 3): This is the location of a point on the directrix, not the focus.
B. (3, 0): This point lies on the x-axis, but the focus must be on the y-axis for this parabola orientation.
D. (-3, 0): Similar to option B, this point lies on the x-axis and is not the correct focus.

Understanding the Parabola Equation

The equation of a parabola provides a mathematical representation of its geometric properties. For a parabola with its vertex at the origin and opening along the y-axis, the general equation takes the form x² = 4py, where p represents the distance between the vertex and the focus (or the vertex and the directrix). In this particular problem, we've established that the directrix is y = 3 and the vertex is at (0, 0). This allows us to determine the value of p and, consequently, the equation of the parabola.

Since the directrix is y = 3, the distance between the vertex (0, 0) and the directrix is 3 units. However, because the parabola opens downwards (as the directrix is above the vertex), the value of p is negative. Thus, p = -3. Now, we can substitute this value into the general equation x² = 4py to obtain the specific equation for this parabola.

Substituting p = -3 into the equation x² = 4py, we get x² = 4(-3)y, which simplifies to x² = -12y. This equation represents the parabola with its vertex at the origin and a directrix of y = 3. It's important to note the negative sign in front of the 12, which indicates that the parabola opens downwards. The equation provides a powerful tool for analyzing and understanding the behavior of the parabola, allowing us to determine various points on the curve and explore its properties.

Conclusion: Mastering Parabola Concepts

In conclusion, determining the focus of a parabola given its vertex and directrix involves a clear understanding of the fundamental properties of parabolas. The definition of a parabola as the locus of points equidistant from the focus and directrix is crucial. Recognizing the role of the vertex as the midpoint between the focus and directrix simplifies the problem. In this case, with the vertex at the origin and the directrix at y = 3, the focus is readily found to be at (0, -3).

The process of analyzing the parabola's orientation and using the distance between the vertex and the directrix is a fundamental skill in conic sections. This understanding not only allows us to solve specific problems but also provides a solid foundation for more advanced topics in mathematics and physics. By mastering the relationship between the vertex, focus, directrix, and the equation of a parabola, we gain a comprehensive understanding of this important geometric shape and its applications in various fields.

This exploration demonstrates how a combination of geometric intuition and algebraic manipulation can lead to a clear and concise solution. The ability to visualize the parabola, understand its properties, and apply the relevant equations is key to success in this type of problem. Mastering these concepts allows for a deeper understanding of parabolas and their role in various mathematical and real-world applications.