To determine the correct properties of the parabola with a given directrix [tex]\( y = 3 \)[/tex], let's analyze the given options and see which one matches the required conditions.
### Step-by-Step Analysis
1. Directrix and Properties:
- The directrix of the parabola is the horizontal line [tex]\( y = 3 \)[/tex].
- For a parabola with a horizontal directrix [tex]\( y = k \)[/tex], the focus is a point [tex]\((h, 2p - k)\)[/tex] and the vertex is equidistant from the directrix and the focus.
2. First Case:
- Focus: [tex]\((0, 3)\)[/tex]
- Equation: [tex]\( y^2 = 12x \)[/tex]
If this were true, the directrix would be [tex]\( y = -3 \)[/tex], because the standard form of the parabola [tex]\( y^2 = 4ax \)[/tex] places the directrix [tex]\( a \)[/tex] units opposite the focus along the y-axis (i.e., [tex]\( k = -3/4 \cdot 12 = -3 \)[/tex]), but our directrix is [tex]\( y = 3 \)[/tex].
Thus, this option is incorrect.
3. Second Case:
- Focus: [tex]\( (0, -3) \)[/tex]
- Equation: [tex]\( x^2 = -12y \)[/tex]
For a vertical parabola, with this equation format, the focus indeed is [tex]\( -p \)[/tex] units from the vertex (here, [tex]\(-3\)[/tex]). Given the provided directrix is correct [tex]\( y = 3 \)[/tex], since the standard form [tex]\( x^2 = -12y \)[/tex] correlates well when the directrix is [tex]\( y \)[/tex]. Specifically, [tex]\( p = -3 \)[/tex] implies a directrix at [tex]\( y = 3 \)[/tex].
Therefore, this option is correct.
4. Third Case:
- Focus: [tex]\( (3, 0) \)[/tex]
- Equation: [tex]\( x^2 = 12y \)[/tex]
For this setup, the directrix must be a vertical line, but our directrix is a horizontal line [tex]\( y = 3 \)[/tex]. This is irrelevant.
Thus, this option is incorrect.
5. Fourth Case:
- Focus: [tex]\( (-3, 0) \)[/tex]
- Equation: [tex]\( y^2 = -12x \)[/tex]
Similarly, the focus here would be invalid for a standard form such a given equation that requires a vertical line directrix. Even equating [tex]\( p \)[/tex], leads to an irrelevant match.
Thus, this option is incorrect.
### Conclusion
The focus is at [tex]\((0, -3)\)[/tex] and the parabola's equation is [tex]\( x^2 = -12y \)[/tex]. Thus, the correct case is:
The focus is at [tex]\((0,-3)\)[/tex], and the equation for the parabola is [tex]\( x^2 = -12y \)[/tex].
