Question

a parabola and its directrix are shown on the graph. what are the coordinates of the focus of the parabola? (0,2) (0,−2) (2,0) (−2,0)

Explanation:

Step1: Recall parabola focus-directrix property

A parabola is equidistant from focus and directrix at any point. The vertex is midway between focus and directrix. The vertex here is at \((0,0)\) (since the parabola opens upward/downward? Wait, the parabola opens upward (since it's a U - shape opening up), and the directrix is \(y = - 2\). The vertex \((h,k)=(0,0)\). The distance from vertex to directrix is \(|k - (-2)|=|0 + 2| = 2\). Since the parabola opens upward, the focus is above the vertex. The y - coordinate of focus is \(k + p\), where \(p\) is the distance from vertex to focus (equal to distance from vertex to directrix). So \(p = 2\), so focus is \((0,0 + 2)=(0,2)\).

Step2: Verify with options

Looking at the options, \((0,2)\) is one of the options.

Answer:

\((0,2)\)