A parabola, with its vertex at $(0,0)$, has a focus on the negative part of the $y$-axis. Which statements about the parabola are true? Select two options.
The directrix will cross through the positive part of the $y$-axis.
The equation of the parabola will be in the form $y^2=4 p x$ where the value of $p$ is negative.
The equation of the parabola will be in the form $x^2=4 p y$ where the value of $p$ is positive.
The equation of the parabola could be $y^2=4 x$.
The equation of the parabola could be $x^2=-\frac{1}{2} y$
Answer (1)
The parabola opens downwards because its focus lies on the negative y-axis.
The directrix of the parabola is on the positive y-axis.
The equation of the parabola is in the form x 2 = 4 p y , where p is negative.
The two correct statements are that the directrix crosses the positive y-axis and the equation could be x 2 = − 2 1 y .
The directrix will cross through the positive part of the y -axis and x 2 = − 2 1 y
Explanation
Determine the direction of the parabola and the location of the directrix. Since the vertex of the parabola is at ( 0 , 0 ) and the focus is on the negative part of the y -axis, the parabola opens downwards. This means the directrix will be on the positive part of the y -axis.
Determine the equation of the parabola. The general equation of a parabola with vertex at the origin and focus on the y -axis is x 2 = 4 p y , where p is the distance from the vertex to the focus. Since the focus is on the negative y -axis, p is negative.
Analyze the given statements. Now, let's analyze the given statements:
The directrix will cross through the positive part of the y -axis. - This is true, as explained in step 1.
The equation of the parabola will be in the form y 2 = 4 p x where the value of p is negative. - This is false. The equation should be in the form x 2 = 4 p y .
The equation of the parabola will be in the form x 2 = 4 p y where the value of p is positive. - This is false. Since the parabola opens downwards, p must be negative.
The equation of the parabola could be y 2 = 4 x . - This is false. The equation should be in the form x 2 = 4 p y .
The equation of the parabola could be x 2 = − 2 1 y - This is true. Here, 4 p = − 2 1 , so p = − 8 1 , which is negative.
State the final answer. Therefore, the two true statements are:
The directrix will cross through the positive part of the y -axis.
The equation of the parabola could be x 2 = − 2 1 y .
Examples
Parabolas are commonly used in the design of satellite dishes. The reflective property of a parabola ensures that signals coming from a satellite, which are parallel to the axis of symmetry, are focused at the focus of the parabola. Knowing the position of the focus is crucial for placing the receiver correctly. In this case, understanding that the directrix is on the positive y-axis and the focus is on the negative y-axis helps in designing the dish efficiently.
