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:

1. For a parabola with vertex at \((0,0)\) and focus on the negative \(y\)-axis:

• The standard form is \(x^{2}=4py\), where \(p\) is the distance from the vertex to the focus (and also from the vertex to the directrix). Since the focus is on the negative \(y\)-axis, \(p < 0\). The directrix is \(y=-p\), and if \(p<0\), then \(-p>0\), so the directrix crosses the positive \(y\)-axis.
• Analyzing each option:
• Option 1: The directrix \(y = - p\) (since \(p<0\), \(-p>0\)) so it crosses the positive \(y\)-axis. This is true.
• Option 2: The parabola opens along the \(y\)-axis, so the equation should be \(x^{2}=4py\), not \(y^{2}=4px\). So this is false.
• Option 3: Since the focus is on the negative \(y\)-axis, \(p<0\), not positive. So this is false.
• Option 4: The equation \(y^{2}=4x\) opens along the \(x\)-axis, but our parabola opens along the \(y\)-axis. So this is false.
• Option 5: The equation \(x^{2}=-\frac{1}{2}y\) can be written as \(x^{2}=4py\) where \(4p=-\frac{1}{2}\), so \(p =-\frac{1}{8}<0\), which matches the focus on the negative \(y\)-axis. This is true.

Answer:

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