The equation of the parabola you provided is x² = 28y. This is a standard form for a parabola that opens upwards, which can be rewritten in the more familiar form of:
(x – h)² = 4p(y – k)
Here, the vertex of the parabola is given by the point (h, k), and 4p is the constant that determines the distance between the vertex and the focus, as well as the distance from the vertex to the directrix.
Finding the Vertex:
In your equation, we can identify the parabola’s vertex:
- Comparing x² = 28y with (x – 0)² = 4p(y – 0), we see that:
- h = 0
- k = 0
Thus, the vertex is at the origin: (0, 0).
Finding the Focus:
To find the focus, we need to determine the value of p. In this equation:
- 4p = 28
- So, p = 28 / 4 = 7.
The focus of the parabola, which lies at a distance of p above the vertex for an upward-opening parabola, is located at:
- (0, 0 + 7) = (0, 7).
Finding the Directrix:
The directrix is a line that is p units below the vertex. Thus, the equation of the directrix is:
- y = k – p = 0 – 7 = -7.
So, the equation of the directrix is y = -7.
Finding the Focal Width:
The focal width of a parabola is the length of the latus rectum, which is also equal to 4p. Since we already found that p = 7, we can calculate:
- Focal Width = 4p = 4 * 7 = 28.
In summary, for the parabola x² = 28y:
- Vertex: (0, 0)
- Focus: (0, 7)
- Directrix: y = -7
- Focal Width: 28
These properties help you understand the geometric significance of the parabola and how it behaves in the coordinate plane.