A parabola with a vertex at $(0,0)$ has a directrix that crosses the negative part of the $y$-axis. Which could be the equation of the parabola?
A. $x^2=-4 y$
B. $x^2=4 y$
C. $y^2=4 x$
D. $y^2=-4 x$
Answer (1)
The parabola has a vertex at (0,0) and a directrix that crosses the negative y-axis.
This means the parabola opens upwards and has the form x 2 = 4 a y , where 0"> a > 0 .
Comparing with the given options, x 2 = 4 y matches this form.
Therefore, the equation of the parabola is x 2 = 4 y .
Explanation
Problem Analysis The problem states that a parabola has its vertex at ( 0 , 0 ) and its directrix intersects the negative y -axis. We need to determine which of the given equations could represent this parabola.
Parabola Orientation Since the directrix intersects the negative y -axis, this means the directrix is a horizontal line of the form y = − a for some 0"> a > 0 . The parabola opens upwards, away from the directrix. The standard form of a parabola with vertex at the origin that opens upwards is x 2 = 4 a y , where a is the distance from the vertex to the focus and from the vertex to the directrix.
Checking the Options Now we examine the given options:
x 2 = − 4 y : This parabola opens downwards, so its directrix intersects the positive y -axis. This option is incorrect.
x 2 = 4 y : This parabola opens upwards, so its directrix intersects the negative y -axis. This option is correct.
y 2 = 4 x : This parabola opens to the right, so its directrix intersects the negative x -axis. This option is incorrect.
y 2 = − 4 x : This parabola opens to the left, so its directrix intersects the positive x -axis. This option is incorrect.
Final Answer Therefore, the equation of the parabola is x 2 = 4 y .
Examples
Parabolas are commonly seen in the real world, such as the curve of a satellite dish or the path of a projectile. Understanding the relationship between the equation of a parabola and its orientation helps engineers design efficient satellite dishes that focus signals onto a receiver. Similarly, understanding parabolic trajectories is crucial in fields like sports and ballistics, where predicting the path of a thrown or fired object is essential.
