To find the vertex, focus, directrix, and focal width of the parabola defined by the equation x = 10y, we need to start by rewriting the equation in standard form.
The equation x = 10y can be rearranged to y = (1/10)x, which represents a parabola that opens to the right.
1. Vertex
The vertex of this parabola is the point where the curve changes direction. For this equation, the vertex is at the origin, which can be represented as:
- Vertex: (0, 0)
2. Focus
The focus of a parabola is a point where light rays converge. The standard form of a parabola that opens to the right is given by:
(y - k)² = 4p(x - h)Here, (h, k) is the vertex and p is the distance from the vertex to the focus. From our equation, we can identify that 4p = 10, hence:
- p = 10/4 = 2.5
Since the parabola opens to the right, the focus is located p units to the right of the vertex:
- Focus: (0 + 2.5, 0) = (2.5, 0)
3. Directrix
The directrix is a line that is perpendicular to the axis of symmetry of the parabola and is located p units to the left of the vertex. Therefore, the equation of the directrix is:
- x = -p = -2.5
4. Focal Width
The focal width of a parabola refers to the distance between the points where the parabola intersects a line that is parallel to the directrix and passes through the focus. For our parabola, the focal width can be calculated using 4p, which we have already identified:
- Focal Width: 4p = 10
Summary:
- Vertex: (0, 0)
- Focus: (2.5, 0)
- Directrix: x = -2.5
- Focal Width: 10
By analyzing the equation of the parabola, we can easily determine its important properties, including the vertex, focus, directrix, and focal width. Understanding these components helps to visualize and graph the parabola more effectively.