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? y² = -2x x² = -2y y² = -8x x² = -8y

Explanation:

Step1: Recall the definition of a parabola

For a parabola, the distance from any point \((x,y)\) on the parabola to the focus \((0, - 2)\) is equal to the distance from the point \((x,y)\) to the directrix \(y = 2\).
The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). The distance from \((x,y)\) to \((0,-2)\) is \(\sqrt{(x - 0)^2+(y + 2)^2}\), and the distance from \((x,y)\) to the line \(y = 2\) is \(|y - 2|\).
So, \(\sqrt{x^{2}+(y + 2)^{2}}=|y - 2|\).

Step2: Square both sides of the equation

$$
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$$

Answer:

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