A parabola has a vertex at the origin. The focus of the parabola is located at ( [tex]$-2,0$[/tex] ).
Which is the equation for the directrix related to the parabola?
A. [tex]$y=2$[/tex]
B. [tex]$x=2$[/tex]
C. [tex]$y=-2$[/tex]
D. [tex]$x=-2$[/tex]
Answer (1)
The parabola has a vertex at (0, 0) and a focus at (-2, 0).
The distance between the vertex and the focus is 2.
The directrix is a vertical line 2 units to the right of the vertex.
The equation of the directrix is x = 2 .
Explanation
Problem Analysis The problem states that a parabola has its vertex at the origin (0, 0) and its focus at (-2, 0). We need to find the equation of the directrix of this parabola.
Understanding the Parabola Since the vertex is at the origin and the focus is at (-2, 0), the parabola opens to the left along the x-axis. The directrix is a vertical line that is equidistant from the vertex as the focus, but on the opposite side of the vertex.
Calculating the Distance The distance between the vertex (0, 0) and the focus (-2, 0) is:
( − 2 − 0 ) 2 + ( 0 − 0 ) 2 = ( − 2 ) 2 = 4 = 2
So, the distance is 2 units.
Determining the Directrix Equation Since the directrix is 2 units away from the vertex on the opposite side of the focus, and the focus is to the left of the vertex, the directrix must be a vertical line 2 units to the right of the vertex. Therefore, the equation of the directrix is x = 2.
Final Answer The equation for the directrix of the parabola is x = 2 .
Examples
Parabolas and their properties, including the directrix, have many real-world applications. For example, satellite dishes and radio telescopes use parabolic reflectors to focus incoming signals onto a single point. The location of the receiver is at the focus, and understanding the directrix helps in optimizing the design and placement of these devices. Similarly, parabolic mirrors in car headlights use the same principle to project a beam of light. In architecture, parabolic arches can provide structural support, and understanding the focus and directrix is crucial for their design.
