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 x^2=-12y. the parabola can be represented by the equation y^2 = 12x.

Explanation:

Step1: Recall parabola properties

For a parabola with vertex at the origin \((0,0)\) and directrix \(y = k\), if \(k>0\), the parabola opens down - ward. The distance from the vertex to the directrix is \(|p|\). Here, the directrix is \(y = 3\), so \(p=- 3\) (negative since the parabola opens down - ward). The focus of a parabola with vertex \((0,0)\) and directrix \(y = k\) is at \((0,-k)\).

Step2: Determine the focus

Since the directrix is \(y = 3\), the focus is at \((0,-3)\) (because the vertex is at the origin and the distance from the vertex to the focus is the same as the distance from the vertex to the directrix).

Step3: Find the equation of the parabola

The standard form of a parabola with vertex \((0,0)\) opening along the y - axis is \(x^{2}=4py\). Substituting \(p = - 3\) into the equation, we get \(x^{2}=4\times(-3)y=-12y\).

Answer:

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