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?
$x^2=-4 y$
$x^2=4 y$
$y^2=4 x$
$y^2=-4 x$
Answer (1)
The parabola has a vertex at (0,0) and its directrix crosses the negative y-axis.
The general form of such a parabola is x 2 = 4 p y with 0"> p > 0 .
Analyzing the given options, x 2 = 4 y satisfies this condition.
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.
General Form of Parabola The general form of a parabola with a vertex at the origin ( 0 , 0 ) is either x 2 = 4 p y or y 2 = 4 p x , where p is the distance from the vertex to the focus and from the vertex to the directrix. The sign of p determines the direction in which the parabola opens.
Parabola Opening Upwards If the directrix intersects the negative y -axis, this means the directrix is of the form y = − p where 0"> p > 0 . In this case, the parabola opens upwards, and its equation is of the form x 2 = 4 p y with 0"> p > 0 .
Examining the Options Now, let's examine the given options:
x 2 = − 4 y : This is of the form x 2 = 4 p y with 4 p = − 4 , so p = − 1 . Since p < 0 , this parabola opens downwards, and its directrix intersects the positive y -axis. Thus, this option is incorrect.
x 2 = 4 y : This is of the form x 2 = 4 p y with 4 p = 4 , so p = 1 . Since 0"> p > 0 , this parabola opens upwards, and its directrix intersects the negative y -axis. Thus, this option is correct.
y 2 = 4 x : This is of the form y 2 = 4 p x with 4 p = 4 , so p = 1 . Since 0"> p > 0 , this parabola opens to the right, and its directrix intersects the negative x -axis. Thus, this option is incorrect.
y 2 = − 4 x : This is of the form y 2 = 4 p x with 4 p = − 4 , so p = − 1 . Since p < 0 , this parabola opens to the left, and its directrix intersects the positive x -axis. Thus, this option is incorrect.
Final Answer Therefore, the equation of the parabola that could have a directrix crossing the negative part of the y -axis is x 2 = 4 y .
Examples
Parabolas are commonly found in the design of satellite dishes and reflecting telescopes. The shape of a parabola allows incoming signals (like radio waves or light) to be focused at a single point, the focus of the parabola. If the directrix of a parabolic reflector is positioned such that it intersects the negative y-axis, it indicates that the reflector is oriented to collect signals from above and concentrate them at the focus, which is a practical application of understanding parabolic geometry.
