Answer:
the equation of the parabola with a focus at (4, -9) and a directrix of y = 1 is (x - 4)^2 = 20(y + 4).
Step-by-step explanation:
To write the equation of the parabola given a focus at (4, -9) and a directrix of y = 1, you can use the standard form of the equation of a parabola with the focus (h, k + p) and directrix y = k - p:
1. Identify the vertex of the parabola:
- The vertex of the parabola is halfway between the focus and the directrix. In this case, the vertex is at (4, (-9 + 1) / 2) = (4, -4).
2. Determine the value of p (the distance between the focus and the vertex):
- p is the distance from the vertex to the focus (which is the same as the distance from the vertex to the directrix). In this case, p = |-9 - (-4)| = 5.
3. Write the equation of the parabola:
- The standard form of the equation of a parabola with the focus (h, k + p) and directrix y = k - p is:
(x - h)^2 = 4p(y - k)
Substituting the values we have:
(x - 4)^2 = 4(5)(y + 4)
(x - 4)^2 = 20(y + 4)
Therefore, the equation of the parabola with a focus at (4, -9) and a directrix of y = 1 is (x - 4)^2 = 20(y + 4).
