Finding The Focus And Directrix Of The Parabola (x-5)^2=8(y-1)

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The world of conic sections is fascinating, and parabolas hold a special place within it. Understanding parabolas is crucial in various fields, from optics and antenna design to understanding projectile motion. To truly grasp a parabola, we need to understand its key components: the focus and the directrix. In this article, we will embark on a step-by-step journey to find the focus and directrix of the given parabola equation: (x-5)^2 = 8(y-1). This process will not only provide the solution but also enhance our comprehension of parabolic equations and their geometric interpretations. By delving into the standard form of parabola equations, we'll unravel the meaning behind each term and how they contribute to defining the shape and position of the parabola. We'll explore the relationship between the focus, directrix, and vertex, which are fundamental in characterizing a parabola. Whether you are a student tackling conic sections for the first time or someone looking to refresh your understanding, this article will equip you with the knowledge and skills to confidently analyze and solve parabola problems. So, let's dive in and demystify the parabola!

Understanding the Standard Form of a Parabola

Before we jump into the specifics of our equation, it’s essential to understand the standard forms of a parabola. Parabolas can open upwards, downwards, leftwards, or rightwards, each having a distinct standard equation. The key to identifying the direction and other parameters lies in recognizing these standard forms. The equation we are dealing with, (x-5)^2 = 8(y-1), resembles the standard form of a parabola that opens either upwards or downwards. This form is given by (x-h)^2 = 4p(y-k), where (h, k) represents the vertex of the parabola, and 'p' is the distance between the vertex and the focus, as well as the distance between the vertex and the directrix. It's important to note that 'p' also determines the direction of the parabola's opening. If 'p' is positive, the parabola opens upwards, and if 'p' is negative, it opens downwards. In contrast, parabolas that open leftwards or rightwards have a standard form of (y-k)^2 = 4p(x-h). The sign of 'p' in this case dictates whether the parabola opens to the right (positive 'p') or to the left (negative 'p'). Recognizing these standard forms is the first step in dissecting any parabolic equation. It allows us to quickly identify the orientation of the parabola and extract the necessary parameters to determine its focus and directrix. Furthermore, understanding the role of the vertex (h, k) is crucial as it serves as the reference point from which we measure the distance to both the focus and the directrix. By grasping these fundamental concepts, we can approach any parabola problem with confidence and clarity.

Identifying the Vertex, Focus, and Directrix

Now, let's apply our understanding of the standard form to our specific equation: (x-5)^2 = 8(y-1). By comparing this equation to the standard form (x-h)^2 = 4p(y-k), we can easily identify the vertex. The vertex (h, k) is the point (5, 1). This is because the equation is already in a form that clearly shows h = 5 and k = 1. The vertex serves as the central point around which the parabola is shaped, and its coordinates are crucial for finding both the focus and the directrix. Next, we need to determine the value of 'p'. In our equation, 8 corresponds to 4p. So, we set 4p = 8 and solve for 'p', which gives us p = 2. This value of 'p' is paramount, as it represents the distance from the vertex to the focus and from the vertex to the directrix. Since the equation is in the form (x-h)^2 = 4p(y-k) and p is positive (p = 2), we know that the parabola opens upwards. This means the focus will be located above the vertex, and the directrix will be a horizontal line below the vertex. To find the coordinates of the focus, we add 'p' to the y-coordinate of the vertex. Therefore, the focus is at (5, 1 + 2) = (5, 3). On the other hand, to find the equation of the directrix, we subtract 'p' from the y-coordinate of the vertex. This gives us the directrix as the line y = 1 - 2, which simplifies to y = -1. Thus, by carefully comparing our equation to the standard form and extracting the values of h, k, and p, we have successfully identified the vertex, focus, and directrix of the parabola. This systematic approach is key to solving any parabola problem and gaining a deeper understanding of these fascinating curves.

Step-by-Step Solution

To solidify our understanding, let's walk through the solution step-by-step, reinforcing each concept along the way.

  1. Identify the Standard Form: The given equation, (x-5)^2 = 8(y-1), is in the form (x-h)^2 = 4p(y-k), which indicates a parabola that opens either upwards or downwards. This is the crucial first step in determining the parabola's orientation and how to proceed with finding the focus and directrix. Recognizing the standard form immediately narrows down our approach and guides us toward the relevant formulas and relationships. It's like having a roadmap before embarking on a journey; it provides direction and clarity.

  2. Determine the Vertex (h, k): By comparing the given equation with the standard form, we can directly identify the vertex. In this case, h = 5 and k = 1, so the vertex is (5, 1). The vertex is the parabola's turning point and serves as the reference from which we measure the distances to the focus and directrix. Its coordinates are fundamental in defining the parabola's position in the coordinate plane. Accurately identifying the vertex is essential for subsequent steps in the solution.

  3. Find the Value of 'p': The term 8 in the equation corresponds to 4p. Setting 4p = 8, we solve for 'p' and find p = 2. The value of 'p' is a critical parameter that determines the distance between the vertex and the focus, as well as the distance between the vertex and the directrix. It also indicates the parabola's opening direction. In this case, since 'p' is positive, the parabola opens upwards.

  4. Locate the Focus: Since the parabola opens upwards, the focus is 'p' units above the vertex. Adding 'p' to the y-coordinate of the vertex, we get the focus at (5, 1 + 2) = (5, 3). The focus is a fixed point inside the parabola, and it has the property that any point on the parabola is equidistant from the focus and the directrix. This property is fundamental to the definition of a parabola and its reflective properties.

  5. Determine the Directrix: The directrix is a horizontal line 'p' units below the vertex. Subtracting 'p' from the y-coordinate of the vertex, we get the directrix as y = 1 - 2, which simplifies to y = -1. The directrix is a fixed line outside the parabola, and it, along with the focus, uniquely defines the parabola. The distance from any point on the parabola to the directrix is equal to its distance to the focus.

By following these steps systematically, we have successfully found the focus and directrix of the parabola (x-5)^2 = 8(y-1). This step-by-step approach not only leads to the correct solution but also reinforces the underlying concepts and relationships within parabolic equations.

Focus: (5, 3)

As we've meticulously calculated, the focus of the parabola (x-5)^2 = 8(y-1) is located at the point (5, 3). The focus, a fundamental element of a parabola, plays a crucial role in defining its shape and properties. It's a fixed point inside the curve, and every point on the parabola maintains an equal distance from the focus and the directrix. This defining characteristic is what gives the parabola its unique reflective property, making it essential in applications like satellite dishes and parabolic microphones. The position of the focus relative to the vertex (in our case, above it since the parabola opens upwards) provides valuable information about the parabola's orientation. Understanding the focus is not just about memorizing coordinates; it's about grasping its significance in the geometric definition of the parabola. It's the linchpin around which the curve is constructed, and its location directly influences the parabola's shape and size. Therefore, accurately determining the focus is paramount in analyzing and applying parabolic equations. In this specific example, the focus (5, 3) tells us that the parabola is vertically oriented, and its opening extends upwards from the vertex (5, 1). This understanding allows us to visualize the parabola's shape and predict its behavior, which is crucial in various real-world applications.

Directrix: y = -1

The directrix of the parabola (x-5)^2 = 8(y-1) is the horizontal line y = -1. The directrix, along with the focus, forms the cornerstone of a parabola's definition. It's a fixed line outside the curve, and like the focus, it dictates the parabola's shape and position. The defining property of a parabola is that every point on the curve is equidistant from the focus and the directrix. This relationship creates the characteristic U-shape of the parabola. In our case, the directrix y = -1 is a horizontal line located below the vertex (5, 1), which is consistent with the parabola opening upwards. The distance between the vertex and the directrix is equal to the distance between the vertex and the focus, a crucial symmetry that governs parabolic geometry. Understanding the directrix is essential for not only solving parabolic equations but also for comprehending the parabola's geometric properties. It provides a reference line against which the parabola's curvature is measured, and its equation gives us a clear visual of the parabola's orientation in the coordinate plane. The directrix is not just a line; it's a fundamental component that, in conjunction with the focus, defines the essence of a parabola.

Conclusion

In this article, we've meticulously dissected the equation (x-5)^2 = 8(y-1) to uncover the focus and directrix of the parabola. We started by establishing a firm understanding of the standard forms of parabolic equations, which provided the framework for our analysis. We then systematically identified the vertex, the value of 'p', and used these parameters to pinpoint the focus and directrix. Through a step-by-step solution, we reinforced each concept and clarified the process of analyzing parabolic equations. We found the focus to be at the point (5, 3) and the directrix to be the line y = -1. These two elements, focus and directrix, are not just coordinates and equations; they are the defining features of the parabola. They dictate its shape, orientation, and reflective properties. By understanding their significance and how to find them, we gain a deeper appreciation for the geometry of parabolas and their applications in various fields. This exercise not only provides the solution to the given problem but also equips us with the knowledge and skills to tackle other parabola-related challenges. The journey through this equation has been a journey through the heart of parabolic geometry, revealing the elegance and precision that lie within these curves.