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 intersecting the negative y-axis.
The general form of such a parabola is x 2 = 4 p y , where 0"> p > 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.
Standard Forms of Parabolas Recall the standard forms of a parabola with vertex at the origin:
If the parabola opens upwards, its equation is of the form x 2 = 4 p y , where 0"> p > 0 and the directrix is y = − p .
If the parabola opens downwards, its equation is of the form x 2 = − 4 p y , where 0"> p > 0 and the directrix is y = p .
If the parabola opens to the right, its equation is of the form y 2 = 4 p x , where 0"> p > 0 and the directrix is x = − p .
If the parabola opens to the left, its equation is of the form y 2 = − 4 p x , where 0"> p > 0 and the directrix is x = p .
Determine the Correct Form Since the directrix intersects the negative y -axis, it is a horizontal line of the form y = − p where 0"> p > 0 . This means the parabola opens upwards. Therefore, the equation of the parabola must be of the form x 2 = 4 p y , where 0"> p > 0 .
Compare with Given Equations Now, let's compare the given equations to the form x 2 = 4 p y :
x 2 = − 4 y : This equation represents a parabola opening downwards, since it is in the form x 2 = − 4 p y with p = 1 . The directrix would be y = 1 , which intersects the positive y -axis. So, this is not the correct equation.
x 2 = 4 y : This equation represents a parabola opening upwards, since it is in the form x 2 = 4 p y with p = 1 . The directrix would be y = − 1 , which intersects the negative y -axis. So, this could be the correct equation.
y 2 = 4 x : This equation represents a parabola opening to the right. The directrix would be x = − 1 , which intersects the negative x -axis. So, this is not the correct equation.
y 2 = − 4 x : This equation represents a parabola opening to the left. The directrix would be x = 1 , which intersects the positive x -axis. So, this is not the correct equation.
Final Answer Therefore, the equation of the parabola that could have a directrix intersecting the negative y -axis is x 2 = 4 y .
Examples
Parabolas are commonly found in the design of satellite dishes and reflecting telescopes. The parabolic shape focuses incoming signals or light to a single point, the focus, where the receiver or eyepiece is placed. The location of the directrix is crucial in determining the shape and focusing properties of the parabola, ensuring optimal signal or image clarity.
