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: Analyze the standard form of parabola

Since the vertex is \((0,0)\) and the focus is on the negative \(y -\)axis, the parabola opens downwards. The standard form of a parabola with vertex \((0,0)\) is \(x^{2}=4py\). When the parabola opens downwards, \(p<0\).

Step2: Analyze the directrix

For a parabola \(x^{2} = 4py\), the directrix is \(y=-p\). Since \(p<0\), \(-p>0\), so the directrix \(y =-p\) crosses the positive \(y -\)axis.

Step3: Check each option

• Option 1:

Since the parabola \(x^{2}=4py\) (\(p <0\)), the directrix \(y=-p\) (\(-p>0\)) crosses the positive \(y -\)axis. This option is True.

• Option 2:

The form \(y^{2}=4px\) is for parabolas that open to the left (\(p<0\)) or to the right (\(p >0\)). Our parabola opens downwards, so this option is False.

• Option 3:

For the parabola \(x^{2}=4py\) that opens downwards, \(p<0\), not \(p>0\). So this option is False.

• Option 4:

The equation \(y^{2}=4x\) is of a parabola that opens to the right (\(p = 1\)). Our parabola opens downwards, so this option is False.

• Option 5:

For the equation \(x^{2}=-\frac{1}{2}y\), we can rewrite it as \(x^{2}=4py\) where \(4p=-\frac{1}{2}\), so \(p=-\frac{1}{8}<0\). This parabola opens downwards. This option is True.

Answer:

The directrix will cross through the positive part of the \(y -\)axis and the equation of the parabola could be \(x^{2}=-\frac{1}{2}y\). So the correct options are:

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