Question

Given the directrix of \(y = 6\) and focus of \((0, 4)\), which is the equation of the parabola?

- A) \(y = -\frac{1}{4}x^2 + 5\)
- B) \(y = -1x^2 - 5\)
- C) \(y = 2x^2 + 5\)
- D) \(y = 1x^2 - 5\)

Answer 1

The equation of the parabola with a directrix of y = 6 and a focus of (0, 4) is A) y = -1/4x^2 + 5, which is determined by the distance from the directrix and focus to the vertex and the parabola's orientation.

Step-by-step explanation:

To determine the equation of a parabola with a given directrix and focus, you can use the definition that a parabola is the set of all points equidistant from the focus and the directrix. In this case, the directrix is y = 6, and the focus is (0, 4). The vertex of the parabola is midway between the directrix and the focus, which is at (0, 5). The distance between the vertex and the focus (or directrix) is known as 'p', which is 1 in this case. The standard form of a parabola that opens up or down is y = a(x-h)^2 + k, where (h, k) is the vertex. Since the focus is below the vertex, the parabola opens downward, hence a is negative. Using p = 1/4a, we find a = -1/4. Plugging in the vertex, we get the equation y = -1/4(x - 0)^2 + 5 which simplifies to y = -1/4x^2 + 5. Therefore, the correct answer is A) y = -1/4x^2 + 5.