A parabola has a vertex at the origin. The equation of the directrix of the parabola is [tex]$y=3$[/tex].
What are the coordinates of its focus?
A. [tex]$(0,3)$[/tex]
B. [tex]$(3,0)$[/tex]
C. [tex]$(0,-3)$[/tex]
D. [tex]$(-3,0)$[/tex]
Answer (1)
The parabola's vertex is at the origin (0, 0) and the directrix is y = 3 .
The focus lies on the y-axis, equidistant from the vertex as the directrix.
The distance from the vertex to the directrix is 3, so the focus is also 3 units from the vertex.
Since the directrix is above the vertex, the focus is below, at ( 0 , − 3 ) .
Explanation
Problem Analysis The problem states that a parabola has its vertex at the origin (0, 0) and the equation of its directrix is given by y = 3 . We need to find the coordinates of the focus of this parabola.
Determining the Focus The vertex of a parabola is equidistant from its focus and its directrix. Since the vertex is at the origin (0, 0) and the directrix is the line y = 3 , the focus must lie on the y-axis. The distance between the vertex and the directrix is ∣3 − 0∣ = 3 . Therefore, the focus must also be a distance of 3 from the vertex, but in the opposite direction from the directrix. Since the directrix is above the vertex, the focus must be below the vertex.
Finding the Coordinates Thus, the coordinates of the focus are (0, -3).
Examples
Parabolas are commonly seen in the real world, such as in the design of satellite dishes and reflecting telescopes. The focus of a parabolic reflector is the point where incoming parallel rays converge after reflection. Understanding the relationship between the vertex, focus, and directrix is crucial for optimizing the design of these devices. For example, if you know the desired location of the focus and the shape of the parabola, you can determine the placement of the directrix to ensure proper functionality.
