Finding The Focus Of A Parabola With Vertex At Origin And Directrix Y=3

In the realm of analytic geometry, parabolas stand out as fundamental conic sections with a rich set of properties and applications. Understanding parabolas is not just an academic exercise; it's crucial for various fields, including physics, engineering, and even art. This article aims to provide a comprehensive explanation of parabolas, focusing on scenarios where the vertex is at the origin and the directrix is a horizontal line. We will dissect the key characteristics of a parabola, particularly the relationship between the vertex, focus, and directrix, and then apply this knowledge to solve the specific problem at hand: determining the coordinates of the focus of a parabola with a vertex at the origin and a directrix given by the equation y = 3.

The parabola is formally 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. 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 point on the parabola that is closest to both the focus and the directrix; it is essentially the "tip" of the parabola. The axis of symmetry is the line that passes through the focus and the vertex, dividing the parabola into two symmetrical halves. This fundamental definition gives rise to the characteristic U-shape of the parabola and dictates the relationships between its key components. For a parabola opening upwards or downwards, the axis of symmetry is a vertical line, and for a parabola opening to the left or right, the axis of symmetry is a horizontal line. The distance between the vertex and the focus is the same as the distance between the vertex and the directrix, a crucial property that helps us determine the equation and key parameters of the parabola.

When the vertex of a parabola is situated at the origin (0,0), the equation of the parabola takes on simplified forms depending on whether it opens upwards, downwards, to the left, or to the right. If the parabola opens upwards, its equation is given by x² = 4py, where p is the distance from the vertex to the focus and also the distance from the vertex to the directrix. The focus in this case is located at (0,p), and the directrix is the line y = -p. Conversely, if the parabola opens downwards, its equation is x² = -4py, with the focus at (0,-p) and the directrix at y = p. For a parabola opening to the right, the equation is y² = 4px, the focus is at (p,0), and the directrix is x = -p. Lastly, if the parabola opens to the left, the equation is y² = -4px, the focus is at (-p,0), and the directrix is x = p. These standard forms are instrumental in quickly identifying the key parameters and orientation of the parabola given its equation or key features. Understanding these relationships allows us to move seamlessly between the geometric definition of a parabola and its algebraic representation.

Now, let's apply this knowledge to the specific problem: a parabola with its vertex at the origin and a directrix given by the equation y = 3. Our goal is to find the coordinates of the focus. Since the vertex is at the origin (0,0) and the directrix is a horizontal line (y = 3), we can infer that the parabola opens downwards. This is because the focus must lie on the opposite side of the vertex from the directrix. The directrix being a horizontal line indicates that the axis of symmetry is vertical, aligning with the y-axis. The distance between the vertex and the directrix is the absolute value of the difference in their y-coordinates, which is |3 - 0| = 3 units. This distance, denoted as p, is also the distance between the vertex and the focus. Therefore, the focus lies 3 units away from the vertex along the axis of symmetry.

Since the parabola opens downwards, the focus will be located below the vertex. Given that the vertex is at (0,0), moving 3 units down along the y-axis brings us to the point (0,-3). Thus, the coordinates of the focus are (0,-3). This result aligns with the standard form equation for a parabola opening downwards, x² = -4py, where p = 3. Substituting p = 3 into the equation gives us x² = -12y, which further confirms that the focus is indeed at (0,-3) and the directrix is at y = 3. This process demonstrates how understanding the relationship between the vertex, focus, and directrix, coupled with the standard forms of parabolic equations, allows us to efficiently determine the key parameters of a parabola.

In summary, for a parabola with a vertex at the origin and a directrix of y = 3, the coordinates of the focus are (0,-3). This conclusion is reached by understanding the fundamental definition of a parabola, recognizing the relationship between the vertex, focus, and directrix, and applying the standard form equation for a parabola opening downwards. This problem underscores the importance of a strong conceptual understanding of conic sections and their properties, which is essential for success in mathematics and related fields. By mastering these concepts, we can confidently tackle more complex problems and appreciate the elegance and utility of parabolas in various applications.

Therefore, the correct answer is C. (0,-3).