Finding The Directrix Of A Parabola Vertex (0,0) Focus (4,0)

When delving into the fascinating world of conic sections, the parabola stands out as a fundamental shape with numerous applications in physics, engineering, and mathematics. A parabola is defined as the set of all points equidistant to a fixed point, known as the focus, and a fixed line, known as the directrix. Understanding the relationship between the vertex, focus, and directrix is crucial for determining the equation of a parabola and its properties. In this article, we will explore how to find the equation of the directrix of a parabola given its vertex and focus. Specifically, we will address the question: A parabola has a vertex at $(0,0)$. The focus of the parabola is located at $(4,0)$. What is the equation of the directrix?

To tackle this problem, we first need to grasp the key components of a parabola. The vertex is the point where the parabola changes direction, and it is the midpoint between the focus and the directrix. The focus is a fixed point that lies inside the curve of the parabola. The directrix is a fixed line that lies outside the curve of the parabola. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. This fundamental property is the essence of the parabola's definition. In our given problem, the vertex is at $(0,0)$ and the focus is at $(4,0)$. Since the vertex is the midpoint between the focus and the directrix, and both the vertex and focus lie on the x-axis, the directrix must be a vertical line. The distance from the vertex to the focus is 4 units. Therefore, the directrix must be 4 units away from the vertex in the opposite direction. This places the directrix at $x = -4$. Thus, the equation of the directrix is $x = -4$.

Key Properties of a Parabola

Before diving deeper into solving the problem, it is essential to review the key properties of a parabola. A parabola is defined as the locus of points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The line passing through the focus and perpendicular to the directrix is called the axis of symmetry, and the point where the parabola intersects its axis of symmetry is the vertex. The distance from the vertex to the focus is the same as the distance from the vertex to the directrix. This distance is often denoted by $p$. The standard form of a parabola's equation depends on whether the parabola opens horizontally or vertically. For a parabola that opens to the right or left, the standard form is $(y - k)^2 = 4p(x - h)$, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus. If $p > 0$, the parabola opens to the right, and if $p < 0$, it opens to the left. For a parabola that opens upwards or downwards, the standard form is $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus. If $p > 0$, the parabola opens upwards, and if $p < 0$, it opens downwards. Understanding these properties and standard forms is crucial for solving problems related to parabolas. In our case, the vertex is at $(0,0)$ and the focus is at $(4,0)$, indicating that the parabola opens to the right. This knowledge helps us determine the equation of the directrix and the overall equation of the parabola.

Step-by-Step Solution to Find the Directrix

To find the equation of the directrix for the given parabola, we will follow a step-by-step approach, ensuring clarity and understanding at each stage. First, we identify the given information: the vertex of the parabola is at $(0,0)$, and the focus is at $(4,0)$. The vertex is a crucial reference point as it lies exactly midway between the focus and the directrix. The focus, being at $(4,0)$, tells us that the parabola opens to the right along the x-axis. This is because the focus lies to the right of the vertex. The distance between the vertex and the focus is a critical parameter. We calculate this distance using the distance formula, which simplifies in this case to the absolute difference in the x-coordinates since both points have the same y-coordinate. The distance $p$ between the vertex $(0,0)$ and the focus $(4,0)$ is $|4 - 0| = 4$ units. The directrix is a line perpendicular to the axis of symmetry (the line passing through the vertex and the focus) and is located on the opposite side of the vertex from the focus. Since the focus is to the right of the vertex, the directrix will be a vertical line to the left of the vertex. The distance from the vertex to the directrix is the same as the distance from the vertex to the focus, which is $p = 4$ units. Therefore, the directrix is a vertical line 4 units to the left of the vertex. Since the vertex is at $x = 0$, the directrix is at $x = 0 - 4 = -4$. Thus, the equation of the directrix is $x = -4$. This step-by-step solution provides a clear methodology for determining the directrix, emphasizing the importance of understanding the geometric properties of a parabola.

Visualizing the Parabola and Directrix

To enhance our understanding, visualizing the parabola, its focus, and directrix can be immensely helpful. Imagine a coordinate plane where the vertex of the parabola is located at the origin $(0,0)$. The focus is at the point $(4,0)$, which lies 4 units to the right of the vertex along the x-axis. The directrix, as we've determined, is a vertical line at $x = -4$, which is 4 units to the left of the vertex. The parabola itself curves around the focus, opening to the right, with the vertex being the point where the curve changes direction. The directrix acts as a boundary that the parabola never crosses. The parabola is the set of all points that are equidistant from the focus $(4,0)$ and the directrix $x = -4$. For example, consider a point on the parabola, say $(4, 4 imes ext{sqrt}(3))$. Its distance from the focus $(4,0)$ can be calculated using the distance formula: $\sqrt{(4-4)^2 + (4\sqrt{3} - 0)^2} = \sqrt{0 + 48} = \sqrt{48} = 4\sqrt{3}$. The distance from the same point to the directrix $x = -4$ is the horizontal distance from the point's x-coordinate to the directrix, which is $|4 - (-4)| = 8$. This example reinforces the definition of a parabola and helps solidify the relationship between the focus, directrix, and the curve itself. Visualizing these components provides a deeper insight into the parabola's properties and the significance of the directrix in its definition.

Understanding the Options

In the given problem, we are presented with four options for the equation of the directrix:

A. $x = -4$ B. $y = -4$ C. $x = 4$ D. $y = 4$

To select the correct answer, we must carefully consider the geometry of the parabola and the roles of the vertex, focus, and directrix. We've already established that the vertex is at $(0,0)$ and the focus is at $(4,0)$. This configuration implies that the parabola opens to the right along the x-axis. The directrix must be a vertical line because it is perpendicular to the axis of symmetry, which is the x-axis in this case. This eliminates options B and D, which represent horizontal lines. The directrix is located on the opposite side of the vertex from the focus, and at the same distance from the vertex. Since the focus is at $x = 4$, the directrix must be at $x = -4$. Therefore, the correct equation of the directrix is $x = -4$, which corresponds to option A. Option C, $x = 4$, would be a vertical line passing through the focus itself, which is not the directrix. By carefully analyzing the properties of the parabola and the given options, we can confidently select the correct equation of the directrix.

Conclusion

In conclusion, determining the equation of the directrix of a parabola involves understanding the fundamental properties of parabolas, particularly the relationship between the vertex, focus, and directrix. Given the vertex at $(0,0)$ and the focus at $(4,0)$, we have systematically shown that the equation of the directrix is $x = -4$. This was achieved by recognizing that the vertex is the midpoint between the focus and the directrix, and that the directrix is a line perpendicular to the axis of symmetry. Visualizing the parabola and its components, and methodically analyzing the given options, further solidified our understanding and led us to the correct solution. This problem exemplifies how a strong grasp of geometric principles and algebraic techniques can effectively solve complex mathematical questions. The ability to find the directrix is not only crucial for understanding the parabola's shape and properties but also for various applications in fields such as optics, engineering, and physics, where parabolic shapes are commonly used to focus or reflect energy and signals.