A parabola has a vertex at the origin. The focus of the parabola is located at ( $-2,0$ ).
Which is the equation for the directrix related to the parabola?
A. $y=2$
B. $x=2$
C. $y=-2$
D. $x=-2$
Answer (2)
The vertex of the parabola is at (0, 0) and the focus is at (-2, 0).
The directrix is a vertical line since the focus and vertex have the same y-coordinate.
The distance between the vertex and the focus is 2 units.
The directrix is located 2 units to the right of the vertex, so its equation is x = 2 .
Explanation
Problem Analysis The problem states that a parabola has a vertex at the origin (0, 0) and its focus is located at (-2, 0). We need to find the equation of the directrix of this parabola.
Determining the Directrix The vertex of a parabola is equidistant from the focus and the directrix. Since the vertex is at (0, 0) and the focus is at (-2, 0), the directrix must be a vertical line (because the focus has the same y-coordinate as the vertex). The distance between the vertex and the focus is the absolute value of the difference in their x-coordinates: ∣ − 2 − 0∣ = 2 . Therefore, the directrix is a vertical line located 2 units to the right of the vertex.
Equation of the Directrix Since the vertex is at the origin (0, 0), and the directrix is a vertical line 2 units to the right, the equation of the directrix is x = 2 .
Examples
Parabolas are commonly used in the design of satellite dishes and reflecting telescopes. The reflective property of a parabola ensures that incoming parallel rays (like radio waves or light) are focused at a single point (the focus). Conversely, a light source placed at the focus will project a parallel beam. Understanding the relationship between the vertex, focus, and directrix is crucial for optimizing the design and performance of these devices.
The equation of the directrix for the parabola with a vertex at the origin and a focus at (-2, 0) is x = 2 . Therefore, the correct answer is B. x = 2 . This reflects the distance and direction of the directrix related to the focus and vertex.
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