The equation of the parabola is given as x² = 12y. This is a standard form of a vertical parabola. To analyze the properties of this parabola, we can identify key components:
1. Vertex
The vertex of a parabola in the form x² = 4py is located at the origin (0, 0). For our equation, we can see:
Here, 4p = 12, so we can solve for p:
p = 12/4 = 3
Thus, the vertex is at the point:
(0, 0)
2. Focus
The focus of a parabola is a point that lies on the axis of symmetry of the parabola and is located at a distance of p from the vertex. Since we found that p = 3, the focus will be at:
(0, 3)
3. Directrix
The directrix is a line that is located p units in the opposite direction from the vertex. For our parabola, the directrix will be:
y = -p = -3
4. Focal Width
The focal width (also known as the latus rectum) is the length of the line segment perpendicular to the axis of symmetry that passes through the focus. For a parabola of the form x² = 4py, the focal width is given by:
Focal Width = 4p
Substituting the value of p:
Focal Width = 4 * 3 = 12
Therefore, the focal width of this parabola is 12.
Summary
In conclusion, for the parabola defined by x² = 12y, we have:
- Vertex: (0, 0)
- Focus: (0, 3)
- Directrix: y = -3
- Focal Width: 12