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 (1)
The vertex of the parabola is at the origin ( 0 , 0 ) and the focus is at ( − 2 , 0 ) .
The directrix is a vertical line x = d .
The x-coordinate of the vertex is the average of the x-coordinate of the focus and the x-value of the directrix: 0 = 2 − 2 + d .
Solving for d , we find d = 2 , so 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 Directrix The directrix is a line such that every point on the parabola is equidistant from the focus and the directrix. Also, the vertex of the parabola is located exactly midway between the focus and the directrix. Since the vertex is at the origin and the focus is at ( − 2 , 0 ) , the directrix must be a vertical line of the form x = d , where d is a constant.
Calculating the Directrix The x-coordinate of the vertex is the average of the x-coordinate of the focus and the x-value of the directrix. Therefore, we have: 0 = 2 − 2 + d Multiplying both sides by 2, we get: 0 = − 2 + d Adding 2 to both sides, we find: d = 2
Final Answer Thus, the equation of the directrix is x = 2 .
Examples
Parabolas are commonly used in the design of satellite dishes and reflecting telescopes. The focus of a parabolic reflector is the point where parallel rays of light or radio waves converge after being reflected by the parabolic surface. The directrix helps define the shape of the parabola, ensuring that all incoming signals are focused correctly. In this case, understanding the relationship between the focus and directrix allows engineers to optimize the design for maximum signal strength.
