To solve the problem, we need to understand the properties of the parabola, given the directrix is the horizontal line [tex]\(y=3\)[/tex].
1. Determine the vertex of the parabola:
- For any parabola, the vertex lies halfway between the focus and the directrix.
- Since the directrix is [tex]\(y = 3\)[/tex], and assuming the vertex lies on the [tex]\(y\)[/tex]-axis, the vertex can be assumed to be at [tex]\((0,0)\)[/tex] without loss of generality.
2. Find the distance from the vertex to the directrix:
- The vertex [tex]\((0,0)\)[/tex] to the directrix [tex]\(y=3\)[/tex] is a distance of 3 units.
3. Determine the position of the focus:
- The focus will be symmetric to the directrix with respect to the vertex.
- Since the directrix is 3 units above the vertex at [tex]\((0,0)\)[/tex], the focus will be the same distance below the vertex.
- Thus, the focus is at [tex]\((0, -3)\)[/tex].
4. Identify the format and equation of the parabola:
- Given a vertex at [tex]\((0,0)\)[/tex] and the focus at [tex]\((0,-3)\)[/tex], the parabola opens downward.
- A downward-opening parabola with the vertex at the origin has the general equation [tex]\(x^2 = -4py\)[/tex], where [tex]\(p\)[/tex] is the distance from the vertex to the focus.
- Here, [tex]\(p = 3\)[/tex], so we get the equation [tex]\(x^2 = -4 \cdot 3 \cdot y = -12y\)[/tex].
So, the correct details about the parabola are:
- The focus is at [tex]\((0, -3)\)[/tex].
- The equation of the parabola is [tex]\(x^2 = -12y\)[/tex].
Thus, the correct option is:
"The focus is at [tex]\((0,-3)\)[/tex], and the equation for the parabola is [tex]\(x^2=-12y\)[/tex]."
