Question

a parabola, with its vertex at the origin, has a directrix at y = 3. which statements about the parabola are true? select two options. the focus is located at (0,−3). the parabola opens to the left. the p value can be determined by computing 4(3). the parabola can be represented by the equation x² = −12y. the parabola can be represented by the equation y² = 12x.

Explanation:

Step1: Recall parabola properties

For a parabola with vertex at the origin \((0,0)\) and directrix \(y = k\), the standard form is \(x^{2}=4p(y - k)\) (opens up/down) or \(y^{2}=4p(x - h)\) (opens left/right). Here, vertex is \((0,0)\), directrix \(y = 3\) (horizontal line), so parabola opens up or down (since directrix is horizontal, axis of symmetry is vertical). The focus is at \((0,p)\) and directrix \(y=-p\) (for vertex at origin, vertical axis). Given directrix \(y = 3\), so \(-p=3\Rightarrow p=- 3\).

Step2: Analyze each option

• Option 1: Focus is \((0,p)=(0, - 3)\). True, since \(p=-3\).
• Option 2: Parabola opens left? No, because axis of symmetry is vertical (directrix horizontal), so it opens up or down (here, since \(p=-3<0\), opens down). So false.
• Option 3: \(p\) value: We found \(p=-3\), not \(4(3)\). False.
• Option 4: Equation: For vertical axis, \(x^{2}=4p y\). With \(p = - 3\), \(x^{2}=4\times(-3)y=-12y\). True.
• Option 5: Equation \(y^{2}=12x\) is for horizontal axis (opens left/right), but our parabola has vertical axis. False.

Answer:

A. The focus is located at \((0, - 3)\)
D. The parabola can be represented by the equation \(x^{2}=-12y\)