To write the equation for a parabola with a focus at (1, 2) and a directrix at y = 6, you can follow these steps:
1. **Understand the Definition**: A parabola is defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
2. **Identify the Focus and Directrix**: The focus is at (1, 2) and the directrix is y = 6.
3. **Find the Vertex**: The vertex of the parabola is the midpoint between the focus and the directrix. Since the directrix is a horizontal line, the y-coordinate of the vertex is the average of the y-coordinate of the focus and the equation of the directrix. So, the vertex is (1, 4).
4. **Determine the Equation**: The standard equation for a parabola with a vertical axis of symmetry and vertex at (h, k) is of the form:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance between the vertex and the focus (also the distance between the vertex and the directrix).
5. **Calculate the Value of p**: The distance between the vertex (1, 4) and the focus (1, 2) is 2 units. Therefore, p = 2.
6. **Substitute the Values into the Equation**: Substituting the vertex (h, k) = (1, 4) and the value of p = 2 into the standard equation gives:
(x - 1)^2 = 8(y - 4)
7. **Final Equation**: The equation for the parabola with a focus at (1, 2) and a directrix at y = 6 is:
(x - 1)^2 = 8(y - 4)
