Finding the focus of a parabola involves identifying key components from the parabola's equation. The process depends on the form of the equation. This response focuses on finding the focus when the equation is in a specific standard form.
Finding the Focus
Here's how to find the focus of a parabola when its equation is in the form (x - h)² = 4a(y - k):
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Identify the Vertex:
- The vertex of the parabola is the point (h, k).
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Determine 'a':
- The value 'a' is found in the equation (x - h)² = 4a(y - k). Solve for 'a'. 'a' represents the directed distance from the vertex to the focus, and from the vertex to the directrix.
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Identify the Axis of the Parabola:
- For the equation (x - h)² = 4a(y - k), the axis of the parabola is parallel to the y-axis. It's a vertical axis passing through the vertex.
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Calculate the Focus:
- Since the axis is parallel to the y-axis, the focus will be 'a' units above or below the vertex. The focus has coordinates (h, k + a). If 'a' is positive, the parabola opens upwards, and the focus is above the vertex. If 'a' is negative, the parabola opens downwards, and the focus is below the vertex.
Example
Let's say we have the equation (x - 2)² = 8(y - 1).
- Vertex: (h, k) = (2, 1)
- 'a' Value: 4a = 8, therefore a = 2.
- Axis: Parallel to the y-axis.
- Focus: (h, k + a) = (2, 1 + 2) = (2, 3)
Therefore, the focus of the parabola (x - 2)² = 8(y - 1) is (2, 3).
Summary Table
| Element | Description |
|---|---|
| Equation Form | (x - h)² = 4a(y - k) |
| Vertex | (h, k) |
| 'a' Value | Distance from vertex to focus/directrix |
| Axis | Parallel to the y-axis |
| Focus | (h, k + a) |
