Question

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 = 4px where the value of p is negative. the equation of the parabola will be in the form x^2 = 4py where the value of p is positive. the equation of the parabola could be y^2 = 4x. the equation of the parabola could be x^2 = -\frac{1}{2}y.

Explanation:

Step1: Recall parabola properties

For a parabola with vertex at $(0,0)$, if the focus is on the negative - part of the $y$-axis, the parabola opens downwards. The standard form of a parabola opening upwards or downwards is $x^{2}=4py$, and if it opens downwards, $p<0$. The directrix of a parabola is a line equidistant from the vertex as the focus, and for a parabola opening downwards, the directrix is a horizontal line above the vertex, so it crosses the positive part of the $y$-axis.

Step2: Analyze each option

• Option 1: Since the focus is on the negative $y$-axis and the directrix is equidistant from the vertex as the focus, the directrix will cross through the positive part of the $y$-axis. This is true.
• Option 2: The form $y^{2}=4px$ represents a parabola that opens to the left or right, not a parabola with focus on the $y$-axis. This is false.
• Option 3: The form $x^{2}=4py$ is correct for a parabola with focus on the $y$-axis, but since the focus is on the negative $y$-axis, $p$ is negative, not positive. This is false.
• Option 4: The form $y^{2}=4x$ represents a parabola opening to the right ($p = 1$), not a parabola with focus on the $y$-axis. This is false.
• Option 5: The equation $x^{2}=-\frac{1}{2}y$ is of the form $x^{2}=4py$ where $4p=-\frac{1}{2}$, so $p =-\frac{1}{8}<0$, which is consistent with a parabola having its focus on the negative $y$-axis. This is true.

Answer:

The directrix will cross through the positive part of the $y$-axis; The equation of the parabola could be $x^{2}=-\frac{1}{2}y$.