Final answer:
The equation of the parabola with the given focus (-5,0) and directrix (y = 2) is y = (1/4)(x+5)² + 5/2, which corresponds to answer option B.
Step-by-step explanation:
To find the equation of a parabola, you need to know the coordinates of its focus and the equation of its directrix. The given focus of the parabola is at (-5,0), and the directrix is the line y = 2. A parabola is the set of all points that are equidistant from the focus and the directrix.
The vertex of the parabola is halfway between the focus and directrix. So, the vertex is at (-5,1), since the distance from the focus at (-5,0) to the directrix y = 2 is 2 units. The vertex form of a parabola's equation is (x - h)² = 4p(y - k), where (h,k) is the vertex and p is the distance from the vertex to the focus or directrix.
Since the directrix is vertical and the focus is horizontal, the parabola opens upwards, so p is positive and its value is 1 (the distance from the vertex (h,k) to the focus (-5,0)). Plugging in the vertex and p into the vertex form, we get (x + 5)² = 4(y - 1). Dividing by 4 and expanding, we get (x + 5)² / 4 = y - 1, and then y = (1/4)(x + 5)² + 1.
Therefore, the equation that represents this parabola is y = (1/4)(x + 5)² + 1, which simplifies to and matches the correct answer B: y = (1/4)(x+5)² + 5/2.