Question

the focus of a parabola is located at (0,−2). the directrix of the parabola is represented by y = 2. which equation represents the parabola?\\(\bigcirc\\ y^2 = -2x\\)\\(\bigcirc\\ x^2 = -2y\\)\\(\bigcirc\\ y^2 = -8x\\)\\(\bigcirc\\ x^2 = -8y\\)

Explanation:

Step1: Determine the vertex and orientation

The vertex of the parabola is halfway between the focus \((0, -2)\) and the directrix \(y = 2\). The \(x\)-coordinate of the vertex is the same as the focus, \(x = 0\). The \(y\)-coordinate is the average of \(-2\) and \(2\), so \(\frac{-2 + 2}{2}=0\). Thus, the vertex is \((0, 0)\). Since the focus is below the vertex and the directrix is above the vertex, the parabola opens downward. For a parabola that opens up or down, the standard form is \(x^{2}=4py\), where \((h, k)\) is the vertex \((0, 0)\) here, so \(h = 0\), \(k = 0\), and \(p\) is the distance from the vertex to the focus (or to the directrix). The distance from the vertex \((0, 0)\) to the focus \((0, -2)\) is \(| - 2-0|=2\), and since it opens downward, \(p=- 2\).

Step2: Substitute \(p\) into the standard form

Substitute \(p=-2\) into the standard form \(x^{2}=4py\). We get \(x^{2}=4\times(-2)y\), which simplifies to \(x^{2}=-8y\).

Answer:

\(x^{2}=-8y\) (the option corresponding to \(x^{2}=-8y\))