Question

Refer to your answers to the questions from Part 1 of Project 1. Any point on the parabola can be labeled (x,y), as shown. A parabola goes through (negative 2, negative 5) & (6, negative 1). A point is above the parabola at (2, negative 4). A line below the parabola goes through (0, negative 6) & (2, negative 6). A point on the parabola is labeled (x, y). © 2019 StrongMind. Created using GeoGebra. What are the distances from the point (x,y) to the focus of the parabola and the directrix?

Answer 1

The distance from a point (x, y) on a parabola to both the focus of the parabola and the directrix is found using the distance formula. The point on the parabola is equidistant from the focus and directrix, and these distances can be calculated using the relevant coordinates for the focus and a perpendicular point on the directrix.

Step-by-step explanation:

The distances from a point (x, y) on a parabola to the focus and the directrix of the parabola are of key importance in understanding the properties of the parabola. Each point on a parabola is equidistant from the focus (a point) and the directrix (a line). Therefore, to find these distances, one would typically use the distance formula d = √((x_2-x_1)^2 + (y_2-y_1)^2) where (x_1, y_1) represents the point on the parabola and (x_2, y_2) represents a point on the focus or the perpendicular point on the directrix, respectively.

To calculate the distance to the focus, you use the coordinates of the focus and the point on the parabola. For the directrix, you determine the vertical distance if the directrix is a horizontal line or use the point-to-line distance formula if the directrix has a different orientation. These calculations are crucial in the study of conic sections, particularly when working with the reflective properties of a parabola such as when analyzing parabolic mirrors or the trajectory of projectiles.