Deriving Parabola Equation Using Focus And Directrix Distances

The parabola, a fundamental conic section, possesses a unique definition rooted in the relationship between a fixed point known as the focus and a fixed line referred to as the directrix. This elegant definition forms the bedrock for deriving the equation of a parabola and understanding its geometric properties. When embarking on the journey to derive the equation of a parabola using the focus and directrix, a crucial step involves equating two distances. Let's dissect the significance of these distances and unravel the underlying concepts. The essence of a parabola lies in its definition: it is the locus of all points that are equidistant from the focus and the directrix. This equidistance property is the cornerstone of the parabola's existence and dictates its characteristic U-shape. The focus, a fixed point denoted as (0, p) in the standard form, exerts a gravitational pull, influencing the curve's direction and curvature. Conversely, the directrix, a fixed line represented by the equation y = -p, acts as a boundary, ensuring that the parabola never crosses it. The interplay between the focus and directrix gives rise to the parabola's distinctive form. The first distance we consider is the distance between a generic point on the parabola, (x, y), and the focus (0, p). This distance, often denoted as d1, embodies the gravitational influence of the focus on the parabola's points. Mathematically, we can express this distance using the distance formula, a direct application of the Pythagorean theorem. The distance formula provides a precise measure of the separation between two points in a coordinate plane, allowing us to quantify the effect of the focus on the parabola's shape. The second distance, equally vital, is the distance between the same generic point on the parabola, (x, y), and the directrix, y = -p. This distance, denoted as d2, represents the boundary imposed by the directrix, preventing the parabola from straying too far. Determining this distance requires a slight adaptation of the distance formula, as we are measuring the separation between a point and a line. The shortest distance between a point and a line is the perpendicular distance, which we can calculate by considering the vertical distance between the point's y-coordinate and the directrix's equation.

The provided equation, $\sqrt{(x-x)^2+(y-(-p))^2}=\sqrt{(x-0)^2+(y-p)^2}$, is a concise mathematical representation of the fundamental principle governing parabolas. It embodies the equality of distances that defines a parabola, forming the foundation for deriving its standard equation. Let's break down this equation piece by piece to gain a deeper understanding. On the left-hand side of the equation, $\sqrt{(x-x)^2+(y-(-p))^2}$, we encounter the distance formula applied to a specific scenario. Notice the terms (x - x) and (y - (-p)). This part of the equation calculates the distance between a point on the parabola, (x, y), and the directrix. It is crucial to recognize that (x-x) simplifies to 0, and (y - (-p)) becomes (y + p). Thus, this side of the equation effectively represents the vertical distance between the point (x, y) and the line y = -p, which is indeed the directrix. The directrix, as we've established, acts as a boundary for the parabola, and this term quantifies how far away a point on the parabola is from this boundary. On the right-hand side of the equation, $\sqrt{(x-0)^2+(y-p)^2}$, we find another application of the distance formula. Here, the terms (x - 0) and (y - p) indicate that we are calculating the distance between the same point on the parabola, (x, y), and the focus, which is located at the point (0, p). The focus, being the defining point around which the parabola curves, plays a pivotal role in shaping the parabola's trajectory. This term measures the gravitational pull of the focus on the points of the parabola. The heart of the equation lies in the equal sign, which signifies that the two distances we've just dissected are, in fact, equal. This equality is the very essence of a parabola's definition: every point on the parabola is equidistant from the focus and the directrix. This equidistance property is not merely a coincidence; it is the defining characteristic that sets parabolas apart from other conic sections. The equation serves as a powerful tool for deriving the standard form of a parabola's equation. By squaring both sides of the equation and simplifying, we can eliminate the square roots and arrive at a more manageable expression. This algebraic manipulation paves the way for isolating the variables and expressing the relationship between x and y in a more transparent manner. The resulting standard equation provides a concise representation of the parabola's shape and position in the coordinate plane.

Given the equation $\sqrt{(x-x)^2+(y-(-p))^2}=\sqrt{(x-0)^2+(y-p)^2}$, we are tasked with identifying what the distance on the left-hand side represents. As we dissected the equation, we identified the components within each square root as applications of the distance formula. The left-hand side, specifically, encapsulates the distance between two key geometric entities. Let's revisit the terms within the square root: (x - x)^2 and (y - (-p))^2. The term (x - x)^2 simplifies to 0, indicating that the x-coordinates are identical. This suggests that we are dealing with a vertical distance, as the horizontal separation is negligible. The term (y - (-p))^2 can be rewritten as (y + p)^2. This term signifies the difference in the y-coordinates. Now, recall that the point (x, y) represents a generic point on the parabola, and the line y = -p represents the directrix. The expression (y + p) effectively calculates the vertical distance between the y-coordinate of the point on the parabola and the directrix. Since the directrix is a horizontal line, the shortest distance between a point and the directrix is indeed the vertical distance. Therefore, the left-hand side of the equation, $\sqrt{(x-x)^2+(y-(-p))^2}$, represents the distance between a point on the parabola and the directrix. This distance is a crucial element in the definition of a parabola, as it is one of the two distances that are set equal to each other. The other distance, represented by the right-hand side of the equation, is the distance between the same point on the parabola and the focus. Equating these two distances is the cornerstone of deriving the parabola's equation. By understanding the geometric interpretation of each side of the equation, we gain a deeper appreciation for the fundamental principles that govern the shape and properties of parabolas. The distance between a point on the parabola and the directrix is not merely a mathematical artifact; it is a tangible representation of the boundary imposed by the directrix, ensuring that the parabola never crosses it. This boundary, in conjunction with the gravitational pull of the focus, dictates the parabola's characteristic U-shape.

Therefore, the answer is:

A. a point on the parabola