Finding The Directrix Of A Parabola Y² = 5x A Step-by-Step Guide

Let's embark on a journey into the fascinating world of parabolas, fundamental shapes in mathematics and physics. Parabolas are more than just curved lines; they are defined by a precise geometric property that relates a point, a line, and a special point known as the focus. To truly grasp the concept of the directrix, we must first solidify our understanding of what defines a parabola. A parabola is defined as the set of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. This definition is the cornerstone of all parabola properties and equations. The focus is a point that lies inside the curve of the parabola, while the directrix is a line that lies outside the curve. The vertex is the turning point of the parabola, lies exactly midway between the focus and the directrix. The axis of symmetry is the line that passes through the focus and the vertex, dividing the parabola into two symmetrical halves. This axis is perpendicular to the directrix. Understanding these basic components is crucial for analyzing and manipulating parabola equations. The relationship between the focus, the directrix, and the points on the parabola is what gives it its unique shape. Every point on the parabola is exactly the same distance from the focus as it is from the directrix. This defining property leads to the standard forms of the parabola equation, which we will explore in more detail later. The direction in which a parabola opens depends on the position of the focus and the directrix. If the focus is to the right of the directrix, the parabola opens to the right. If the focus is to the left of the directrix, the parabola opens to the left. Similarly, if the focus is above the directrix, the parabola opens upwards, and if the focus is below the directrix, it opens downwards. This directional property is directly reflected in the equation of the parabola. For instance, an equation in the form y² = 4ax represents a parabola that opens to the right, while an equation in the form x² = 4ay represents a parabola that opens upwards. By understanding these fundamental properties, we can analyze and interpret parabola equations, and more importantly, we can determine the equation of the directrix, which is the core of our discussion.

Now, let's dive into the specific equation at hand: y² = 5x. This equation represents a parabola in standard form, but to fully understand it, we need to relate it to the general forms of parabola equations. This standard form provides immediate insights into the parabola's orientation and key parameters. By comparing y² = 5x to the standard forms, we can extract the information needed to find the directrix. The general form for a parabola opening to the right is y² = 4ax, where 'a' is the distance between the vertex and the focus, and also the distance between the vertex and the directrix. Recognizing this form allows us to quickly identify the key parameter 'a' in our given equation. In our case, y² = 5x, we can equate 4a to 5. This simple equation, 4a = 5, is the key to unlocking the parabola's properties. Solving for 'a' will give us the crucial distance that separates the vertex from both the focus and the directrix. Once we determine the value of 'a', we can pinpoint the exact location of the directrix. Solving 4a = 5, we find that a = 5/4. This value represents the distance between the vertex and the focus, as well as the distance between the vertex and the directrix. It's a critical piece of information for determining the equation of the directrix. The vertex of this parabola is at the origin (0,0). Since the equation is in the form y² = 4ax, the parabola opens to the right. The focus will be located at a distance of 'a' to the right of the vertex, and the directrix will be located at a distance of 'a' to the left of the vertex. Knowing that the vertex is at (0,0) and a = 5/4, we can deduce the coordinates of the focus and the equation of the directrix. The focus will be at (5/4, 0), and the directrix will be a vertical line located 5/4 units to the left of the vertex. This visualization helps to solidify our understanding of the parabola's geometry and the role of the directrix in defining its shape. By carefully comparing the given equation to the standard form, we have successfully extracted the value of 'a', which is the cornerstone for determining the equation of the directrix. The next step is to use this information to write the equation of the directrix, which will be one of the answer choices provided.

With a = 5/4 determined, we are now poised to find the equation of the directrix. The directrix, as we know, is a line located 'a' units away from the vertex, on the opposite side of the focus. Since our parabola y² = 5x opens to the right and its vertex is at the origin (0,0), the directrix will be a vertical line to the left of the y-axis. This is because the focus lies to the right of the vertex for this type of parabola. Understanding the orientation of the parabola and the position of the vertex is crucial for correctly determining the equation of the directrix. If the parabola opened to the left, the directrix would be to the right of the vertex. If it opened upwards or downwards, the directrix would be a horizontal line. Since the directrix is a vertical line, its equation will be in the form x = constant. The constant value represents the x-coordinate where the line intersects the x-axis. We know the directrix is located 'a' units to the left of the vertex, and a = 5/4. Therefore, the x-coordinate of the directrix is -5/4. The equation of the directrix is thus x = -5/4. This equation represents a vertical line passing through the point (-5/4, 0) on the x-axis. It is the line from which all points on the parabola are equidistant to the focus. The equation x = -5/4 perfectly describes the directrix of the parabola y² = 5x. It encapsulates the geometric relationship between the parabola, its vertex, and the defining line that shapes its curve. Now, let's compare our result, x = -5/4, with the answer choices provided. This comparison will confirm our solution and lead us to the correct answer. The process of finding the directrix involves understanding the parabola's equation, determining the value of 'a', and applying the geometric properties of the directrix relative to the vertex. This step-by-step approach ensures accuracy and reinforces the fundamental concepts of parabola geometry.

By comparing our derived equation, x = -5/4, with the given options, we can confidently identify the correct answer. The options were:

A. y = -20 B. x = -20 C. y = -5/4 D. x = -5/4

Our solution, x = -5/4, matches option D. Therefore, the correct answer is D. x = -5/4. This concludes our detailed exploration of the parabola y² = 5x and its directrix. We have successfully navigated the process of identifying the equation of the directrix by understanding the fundamental properties of parabolas, relating the given equation to the standard form, and applying geometric principles. This comprehensive approach not only provides the correct answer but also deepens our understanding of conic sections and their defining characteristics. The directrix is a crucial element in defining a parabola, and understanding its relationship to the focus and vertex is essential for analyzing and manipulating parabola equations. Our journey through this problem has highlighted the importance of a systematic approach, from understanding the basic definitions to applying them in a specific context. By carefully examining the equation, extracting key parameters, and applying the geometric properties, we have confidently arrived at the correct solution. This process reinforces the power of mathematical reasoning and the importance of a solid foundation in fundamental concepts. The equation of the directrix, x = -5/4, is not just a number; it's a geometric descriptor that defines the shape and position of the parabola in the coordinate plane. Understanding this connection between equations and geometric shapes is a key aspect of mathematical proficiency. As we conclude this exploration, we recognize that the ability to find the directrix of a parabola is a valuable skill in mathematics and its applications. It demonstrates a deep understanding of conic sections and their properties, paving the way for further exploration of more complex mathematical concepts.