To analyze the parabola given by the equation x = 10y, we can rewrite it in the standard form of a parabola.
This equation can be rearranged to:
y = (1/10)x
This indicates that it opens to the right, and it is in the form of x = 4py, where p is the distance from the vertex to the focus. From the equation x = 10y, we can identify that:
- 4p = 10
- p = 10/4 = 2.5
Step 1: Finding the Vertex
The vertex of the parabola is at the origin (0, 0).
Step 2: Finding the Focus
The focus is located at a distance of p units from the vertex along the axis of symmetry. For our parabola:
- Focus: (p, 0) = (2.5, 0)
Step 3: Finding the Directrix
The directrix is a line that is p units in the opposite direction from the vertex along the axis of symmetry. Thus, we find:
- Directrix: x = -p = -2.5
Step 4: Finding the Focal Width
The focal width is the distance across the parabola at the focus and is equal to |4p|. So:
- Focal Width = |4p| = |10| = 10
Summary:
- Vertex: (0, 0)
- Focus: (2.5, 0)
- Directrix: x = -2.5
- Focal Width: 10
This detailed breakdown helps in understanding the components of the parabola clearly.