Finding The Focus And Directrix Of A Parabola (y-6)^2=12(x-2)

In this article, we will delve into the fascinating world of parabolas and explore how to determine their key features: the focus and the directrix. A parabola is a conic section defined as the set of all points equidistant to a fixed point (the focus) and a fixed line (the directrix). Understanding these elements is crucial for grasping the geometry and applications of parabolas in various fields, from optics and antennas to trajectory analysis and engineering. Our specific task is to find the focus and directrix of the parabola given by the equation $(y-6)^2=12(x-2)$. This equation is in a standard form that allows us to readily identify the vertex, axis of symmetry, and the distance between the vertex and the focus (and the vertex and the directrix). By carefully analyzing the equation and applying the relevant formulas, we will pinpoint the coordinates of the focus and derive the equation of the directrix. This process will not only provide a solution to the given problem but also enhance our overall understanding of parabolas and their properties. We will break down the problem step-by-step, providing clear explanations and justifications for each step, making it accessible to anyone interested in learning more about conic sections. So, let’s embark on this mathematical journey and unravel the mysteries of the parabola $(y-6)^2=12(x-2)$.

Understanding the Standard Form of a Parabola

The key to finding the focus and directrix of a parabola lies in recognizing and understanding its standard form equation. Parabolas can open either horizontally or vertically, and each orientation has its own standard form. For a parabola that opens horizontally, the standard form equation is given by $(y-k)^2 = 4p(x-h)$, 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. The sign of $p$ determines the direction in which the parabola opens; if $p$ is positive, the parabola opens to the right, and if $p$ is negative, it opens to the left. Similarly, for a parabola that opens vertically, the standard form equation is given by $(x-h)^2 = 4p(y-k)$, where $(h, k)$ is the vertex, and $p$ is the distance between the vertex and the focus (and the vertex and the directrix). In this case, if $p$ is positive, the parabola opens upwards, and if $p$ is negative, it opens downwards. Understanding these standard forms allows us to quickly identify the vertex and the value of $p$, which are essential for determining the focus and directrix. The vertex $(h, k)$ is the turning point of the parabola, and it serves as a reference point for locating the focus and directrix. The value of $p$ not only gives us the distance but also the orientation of the parabola, helping us visualize its shape and position in the coordinate plane. By mastering the standard forms, we can efficiently analyze and solve problems involving parabolas, making them less daunting and more accessible.

Identifying the Vertex, Orientation, and Value of p

To effectively analyze the given parabola equation, $(y-6)^2=12(x-2)$, we need to identify its key parameters. The first step is to recognize that the equation is in the standard form $(y-k)^2 = 4p(x-h)$, which indicates that the parabola opens horizontally. This is because the $y$ term is squared, and the $x$ term is linear. By comparing the given equation with the standard form, we can readily identify the vertex of the parabola. The vertex $(h, k)$ corresponds to the values that make the squared terms zero. In our equation, $(y-6)^2$ becomes zero when $y=6$, and $(x-2)$ becomes zero when $x=2$. Therefore, the vertex of the parabola is $(2, 6)$. Next, we need to determine the value of $p$, which represents the distance between the vertex and the focus, as well as the distance between the vertex and the directrix. In the standard form equation, the coefficient of the linear term is $4p$. In our given equation, $12$ corresponds to $4p$. Thus, we can set up the equation $4p = 12$ and solve for $p$. Dividing both sides by $4$, we find that $p = 3$. Since $p$ is positive, this confirms that the parabola opens to the right. The value of $p$ is crucial because it dictates the position of the focus and the directrix relative to the vertex. With the vertex $(2, 6)$ and $p = 3$ identified, we have the necessary information to calculate the coordinates of the focus and the equation of the directrix. This systematic approach of comparing the given equation to the standard form and extracting the relevant parameters is fundamental to solving parabola problems and understanding their geometric properties.

Determining the Focus of the Parabola

With the vertex and the value of $p$ determined, we can now proceed to find the focus of the parabola. Recall that the focus is a point inside the curve of the parabola, and its distance from the vertex is given by $|p|$. Since our parabola has the equation $(y-6)^2=12(x-2)$, which is in the standard form $(y-k)^2 = 4p(x-h)$, we know that it opens horizontally. Furthermore, since $p = 3$ is positive, the parabola opens to the right. The focus of a horizontally oriented parabola is located along the axis of symmetry, which in this case is the horizontal line $y = 6$ (since the vertex is $(2, 6)$). To find the coordinates of the focus, we need to move a distance of $p$ units from the vertex along the axis of symmetry in the direction the parabola opens. Since the parabola opens to the right, we will move $p = 3$ units to the right of the vertex. The $x$-coordinate of the vertex is $2$, so we add $p$ to it: $2 + 3 = 5$. The $y$-coordinate of the focus remains the same as the $y$-coordinate of the vertex, which is $6$. Therefore, the focus of the parabola is $(5, 6)$. This means that all points on the parabola are equidistant to this point and to the directrix. Understanding how the value of $p$ and the orientation of the parabola determine the position of the focus is essential for accurately solving parabola problems. By systematically applying the standard form equations and the definition of the focus, we can confidently locate this key feature of the parabola.

Finding the Directrix of the Parabola

After locating the focus, the next step is to determine the directrix of the parabola. The directrix is a line that lies outside the curve of the parabola, and it is also a distance of $|p|$ from the vertex, but in the opposite direction from the focus. For the parabola given by the equation $(y-6)^2=12(x-2)$, we have already established that the vertex is $(2, 6)$, $p = 3$, and the parabola opens to the right. Since the parabola opens horizontally, the directrix will be a vertical line. To find the equation of the directrix, we need to move a distance of $p = 3$ units from the vertex along the axis of symmetry, but in the opposite direction from the focus. Since the focus is to the right of the vertex, the directrix will be to the left of the vertex. The $x$-coordinate of the vertex is $2$, so we subtract $p$ from it: $2 - 3 = -1$. This gives us the $x$-coordinate of the vertical line that represents the directrix. The equation of a vertical line is given by $x = c$, where $c$ is a constant. In this case, the directrix is the vertical line $x = -1$. This means that any point on the parabola is equidistant from the focus $(5, 6)$ and the directrix $x = -1$. The directrix plays a crucial role in the definition of a parabola, and understanding its relationship to the vertex and focus is essential for a complete understanding of parabolas. By correctly applying the standard form equations and the definition of the directrix, we can accurately determine its equation and gain a deeper insight into the geometry of parabolas.

Conclusion

In conclusion, we have successfully found the focus and directrix of the parabola given by the equation $(y-6)^2=12(x-2)$. By comparing the given equation to the standard form $(y-k)^2 = 4p(x-h)$, we identified the vertex as $(2, 6)$ and determined that $p = 3$. This allowed us to deduce that the parabola opens to the right. Using the value of $p$ and the position of the vertex, we calculated the coordinates of the focus, which is $(5, 6)$. We also found the equation of the directrix, which is the vertical line $x = -1$. Throughout this process, we emphasized the importance of understanding the standard form of a parabola and how it relates to the vertex, focus, and directrix. We also highlighted the significance of the value of $p$ in determining the distance between the vertex and the focus, as well as the distance between the vertex and the directrix. By systematically applying these concepts, we were able to solve the problem and gain a deeper understanding of the properties of parabolas. This knowledge is not only valuable for solving mathematical problems but also for understanding the applications of parabolas in various fields such as physics, engineering, and computer graphics. The ability to find the focus and directrix of a parabola is a fundamental skill in the study of conic sections, and mastering this skill opens the door to more advanced topics in mathematics and its applications. We hope this detailed explanation has provided a clear and comprehensive understanding of how to find the focus and directrix of a parabola, empowering you to tackle similar problems with confidence.