To solve the problem, let's follow a step-by-step approach:
1. Understanding the Vertex and Directrix:
- The given parabola has its vertex at the origin, which is [tex]\((0, 0)\)[/tex].
- The directrix of the parabola is given as [tex]\(y = 3\)[/tex].
2. Identifying the Direction of Opening:
- Given that the directrix ([tex]\(y = 3\)[/tex]) is above the vertex ([tex]\(0, 0\)[/tex]), the parabola must open downwards.
3. Determining the Parameters of the Parabola:
- For a parabola that opens vertically (either upwards or downwards), the general form of the equation is [tex]\((x - h)^2 = 4p(y - k)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex and [tex]\(p\)[/tex] is the distance from the vertex to the focus (and also the distance from the vertex to the directrix, but in the opposite direction).
- Here, the vertex is at [tex]\((0, 0)\)[/tex], so [tex]\(h = 0\)[/tex] and [tex]\(k = 0\)[/tex].
- The directrix is at [tex]\(y = 3\)[/tex]. The distance [tex]\(p\)[/tex] from the vertex to the directrix is thus [tex]\(p = 3\)[/tex].
4. Focus of the Parabola:
- Since the parabola opens downwards, the focus lies below the vertex at a distance of [tex]\(p = 3\)[/tex].
- Therefore, the focus of the parabola is at [tex]\((0, -3)\)[/tex].
5. Equation of the Parabola:
- Substituting [tex]\(p = 3\)[/tex] into the general form equation:
[tex]\[
(x - 0)^2 = 4 \cdot 3 \cdot (y - 0) \implies x^2 = 12y
\][/tex]
- Since the parabola opens downwards, the equation should reflect this by having a negative value:
[tex]\[
x^2 = -12y
\][/tex]
By following these steps, we can conclude that the two correct statements about the parabola are:
1. The focus is located at [tex]\((0, -3)\)[/tex].
2. The parabola can be represented by the equation [tex]\(x^2 = -12y\)[/tex].
Thus, our final answer is:
- The focus is located at (0, -3).
- The parabola can be represented by the equation [tex]\(x^2 = -12 y\)[/tex].
