Parabola Equation From Focus And Directrix A Step-by-Step Guide

Determining the equation of a parabola when given its focus and directrix is a fundamental concept in mathematics, particularly in analytic geometry. This article aims to provide a detailed explanation of how to derive the equation of a parabola given its focus at $(-1, 15)$ and directrix $x = -4$. We will explore the underlying principles, the standard form of a parabola equation, and the step-by-step process to arrive at the correct solution. Whether you're a student grappling with conic sections or a math enthusiast looking to deepen your understanding, this guide will offer valuable insights and clarity. Let's dive into the world of parabolas and unravel the mystery behind their equations.

Defining 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 crucial in understanding how to derive the equation of a parabola. The axis of symmetry is the line that passes through the focus and is perpendicular to the directrix. The vertex is the point where the parabola intersects its axis of symmetry, and it is the midpoint between the focus and the directrix. These elements—focus, directrix, axis of symmetry, and vertex—are essential in constructing the equation of a parabola.

Understanding the relationship between these elements is key to solving problems involving parabolas. For instance, the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix. This property is not just a theoretical concept but a practical tool that we use to derive the equation. By applying the distance formula and setting the distances equal to each other, we can create an equation that describes the parabola. This equation is a powerful representation of the geometric shape and allows us to analyze its properties and behavior.

Moreover, the orientation of the parabola depends on the position of the focus and the directrix. If the directrix is a vertical line, the parabola opens either to the left or to the right. If the directrix is a horizontal line, the parabola opens either upwards or downwards. The focus always lies inside the curve of the parabola, while the directrix is always outside. Recognizing these relationships helps in visualizing the parabola and predicting the form of its equation even before performing any calculations. This holistic understanding is what makes the study of parabolas both challenging and rewarding.

Standard Forms of Parabola Equations

The equation of a parabola can take different forms depending on its orientation. For a parabola that opens horizontally (left or right), the standard form is:

$$(y - k)^2 = 4p(x - h)$$

where $(h, k)$ is the vertex of the parabola, and $p$ is the distance from the vertex to the focus and from the vertex to the directrix. If $p > 0$, the parabola opens to the right, and if $p < 0$, it opens to the left.

For a parabola that opens vertically (upwards or downwards), the standard form is:

$$(x - h)^2 = 4p(y - k)$$

Again, $(h, k)$ is the vertex, and $p$ is the distance from the vertex to the focus and from the vertex to the directrix. If $p > 0$, the parabola opens upwards, and if $p < 0$, it opens downwards.

These standard forms are invaluable tools for finding the equation of a parabola. They provide a structured way to incorporate the key parameters—the vertex and the distance $p$—into the equation. Recognizing which form to use based on the orientation of the parabola is a crucial step in the problem-solving process. For example, if the directrix is a vertical line, the parabola will open horizontally, and the first standard form should be used. Conversely, if the directrix is a horizontal line, the parabola will open vertically, and the second standard form is appropriate.

Understanding the significance of the parameter $p$ is equally important. It not only determines the direction in which the parabola opens but also affects its "width." A larger value of $|p|$ means the parabola is wider, while a smaller value means it is narrower. This parameter is directly related to the curvature of the parabola and plays a vital role in various applications, such as the design of parabolic reflectors used in satellite dishes and telescopes. By mastering these standard forms and the parameters they involve, one can effectively analyze and manipulate parabola equations in a variety of contexts.

Determining the Vertex

Given the focus $(-1, 15)$ and the directrix $x = -4$, we can find the vertex of the parabola. The vertex is the midpoint between the focus and the directrix. Since the directrix is a vertical line, the parabola opens horizontally. The x-coordinate of the vertex is the average of the x-coordinate of the focus and the x-value of the directrix.

h = rac{-1 + (-4)}{2} = rac{-5}{2}

The y-coordinate of the vertex is the same as the y-coordinate of the focus, which is 15. Therefore, the vertex is (-rac{5}{2}, 15).

Finding the vertex is a pivotal step in determining the equation of a parabola. It provides the center around which the parabola is symmetric and serves as a reference point for constructing the equation. The method of finding the vertex as the midpoint between the focus and the directrix is a direct application of the definition of a parabola. This approach is not only straightforward but also conceptually sound, reinforcing the geometric interpretation of the parabola.

In this specific case, recognizing that the directrix is a vertical line immediately tells us that the parabola opens either to the right or to the left. This understanding guides us in choosing the appropriate standard form of the equation. The calculation of the x-coordinate of the vertex involves a simple averaging process, which is easy to implement. The y-coordinate remains unchanged because the axis of symmetry is a horizontal line passing through the focus. The resulting vertex coordinates, (-rac{5}{2}, 15), now provide crucial information for the next steps in deriving the parabola's equation. This methodical approach, breaking down the problem into manageable steps, is essential for solving more complex problems in analytic geometry.

Calculating the Distance 'p'

The distance $p$ is the distance between the vertex and the focus (or the vertex and the directrix). This distance is crucial for determining the shape and orientation of the parabola. In our case, the vertex is (-rac{5}{2}, 15) and the focus is $(-1, 15)$.

p = -1 - (-rac{5}{2}) = -1 + rac{5}{2} = rac{3}{2}

Since $p$ is positive, the parabola opens to the right.

The distance $p$ is a critical parameter that encapsulates the fundamental relationship between the vertex, focus, and directrix of a parabola. It is not just a numerical value but a geometric measure that dictates the parabola's curvature and direction. The calculation of $p$ involves finding the directed distance between the vertex and the focus, which in this context is a straightforward subtraction of the x-coordinates, since the y-coordinates are the same. The result, p = rac{3}{2}, provides immediate insight into the parabola's orientation.

The fact that $p$ is positive in this case is significant. It confirms that the parabola opens to the right, which aligns with the understanding that the focus lies to the right of the directrix. This directional information is essential for selecting the correct form of the parabola equation and for visualizing the parabola's shape. The magnitude of $p$ also gives an indication of how "wide" or "narrow" the parabola is; a larger $p$ would result in a wider parabola, while a smaller $p$ would lead to a narrower one. This parameter is therefore central to both the algebraic representation and the geometric interpretation of the parabola.

Furthermore, the calculation of $p$ underscores the importance of understanding the signs and directions in coordinate geometry. The directed distance considers the order of the points, which is crucial for determining the correct sign of $p$. This careful attention to detail is a hallmark of precise mathematical reasoning and is essential for avoiding errors in problem-solving. The calculated value of $p$ now serves as a key ingredient in the final step of constructing the parabola's equation, bridging the geometric properties with the algebraic formulation.

Constructing the Equation

Since the parabola opens to the right, we use the standard form:

$$(y - k)^2 = 4p(x - h)$$

where (h, k) = (-rac{5}{2}, 15) and p = rac{3}{2}.

Substituting these values, we get:

(y - 15)^2 = 4(rac{3}{2})(x - (-rac{5}{2}))

(y - 15)^2 = 6(x + rac{5}{2})

To express $x$ in terms of $y$, we rearrange the equation:

x + rac{5}{2} = rac{1}{6}(y - 15)^2

x = rac{1}{6}(y - 15)^2 - rac{5}{2}

Thus, the equation of the parabola is:

x = rac{1}{6}(y - 15)^2 - rac{5}{2}

Constructing the equation of the parabola is the culmination of the preceding steps, bringing together the vertex coordinates and the distance $p$ into a cohesive algebraic representation. The choice of the standard form $(y - k)^2 = 4p(x - h)$ is dictated by the fact that the parabola opens horizontally, as determined by the position of the directrix. This selection is a critical juncture, as using the incorrect form would lead to an entirely different equation.

The substitution of the vertex coordinates (h, k) = (-rac{5}{2}, 15) and the value p = rac{3}{2} is a direct application of the standard form. The arithmetic involved in this substitution must be meticulously executed to avoid errors. The simplification process, from (y - 15)^2 = 6(x + rac{5}{2}) to x = rac{1}{6}(y - 15)^2 - rac{5}{2}, involves algebraic manipulation to isolate $x$ on one side of the equation. This rearrangement not only presents the equation in a more conventional form but also facilitates the analysis of the parabola's properties.

The final equation, x = rac{1}{6}(y - 15)^2 - rac{5}{2}, is a complete algebraic description of the parabola. It encapsulates all the geometric information derived from the focus and directrix. This equation can be used to plot the parabola, determine its points, and analyze its behavior. The process of arriving at this equation exemplifies the power of analytic geometry in translating geometric concepts into algebraic expressions and vice versa. This final step underscores the interconnectedness of the various elements of the parabola and highlights the elegance of mathematical reasoning.

Conclusion

Given the focus $(-1, 15)$ and the directrix $x = -4$, the equation of the parabola is:

x = rac{1}{6}(y - 15)^2 - rac{5}{2}

This corresponds to option A.

In conclusion, determining the equation of a parabola from its focus and directrix is a multi-faceted process that involves understanding the fundamental definition of a parabola, recognizing the standard forms of parabola equations, and applying algebraic techniques to derive the specific equation. The journey from the geometric properties—the focus and directrix—to the algebraic representation—the equation—is a testament to the power of analytic geometry.

Each step in this process, from identifying the vertex to calculating the distance $p$ and constructing the equation, is crucial. A thorough understanding of these steps not only enables the solution of specific problems but also fosters a deeper appreciation for the underlying principles of conic sections. The ability to translate between geometric and algebraic representations is a valuable skill in mathematics, with applications spanning various fields, including physics, engineering, and computer graphics.

This article has provided a comprehensive guide to solving this particular problem, but the techniques and concepts discussed are broadly applicable to a wide range of parabola-related problems. By mastering these skills, students and enthusiasts alike can confidently tackle more complex challenges and further explore the rich landscape of mathematical concepts. The final equation, x = rac{1}{6}(y - 15)^2 - rac{5}{2}, stands as a concise and elegant summary of the parabola's characteristics, encapsulating its position, orientation, and shape in a single algebraic expression.