To determine the equation of the parabola given the focus [tex]\((-1, 15)\)[/tex] and the directrix [tex]\(x = -4\)[/tex], let's go through the steps systematically.
### Step 1: Find the coordinates of the vertex
The vertex of a parabola with a horizontal directrix is the midpoint between the focus and the directrix.
- Focus: [tex]\((-1, 15)\)[/tex]
- Directrix: [tex]\(x = -4\)[/tex]
Calculate the x-coordinate of the vertex:
[tex]\[
\text{Vertex}_x = \frac{\text{Focus}_x + \text{Directrix}_x}{2} = \frac{-1 + (-4)}{2} = \frac{-5}{2} = -2.5
\][/tex]
The y-coordinate of the vertex is the same as the y-coordinate of the focus, which is 15.
- Vertex: [tex]\((-2.5, 15)\)[/tex]
### Step 2: Calculate the distance between the focus and the directrix
This distance is used to find the value of [tex]\(p\)[/tex], which represents the distance from the vertex to the focus (or the vertex to the directrix, as both are the same).
[tex]\[
\text{Distance} = \left| \text{Focus}_x - \text{Directrix}_x \right| = \left| -1 - (-4) \right| = \left| -1 + 4 \right| = 3
\][/tex]
The distance in terms of [tex]\(p\)[/tex]:
[tex]\[
p = \frac{\text{Distance}}{2} = \frac{3}{2} = 1.5
\][/tex]
### Step 3: Formulate the equation of the parabola
For a parabola with a horizontal directrix, the general equation is:
[tex]\[
(x - h) = \frac{1}{4p}(y - k)^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Given [tex]\(h = -2.5\)[/tex], [tex]\(k = 15\)[/tex], and [tex]\(p = 1.5\)[/tex], substitute these into the equation:
[tex]\[
x + 2.5 = \frac{1}{4 \times 1.5}(y - 15)^2 = \frac{1}{6}(y - 15)^2
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1}{6}(y - 15)^2 - 2.5
\][/tex]
### Step 4: Adjust for direction
Since the focus is to the right of the directrix [tex]\(x = -4\)[/tex], the parabola opens to the left. Thus, the [tex]\( \frac{1}{6} \)[/tex] should be negative in sign:
[tex]\[
x = -\frac{1}{6}(y - 15)^2 - 2.5
\][/tex]
### Selecting the correct answer:
D. [tex]\(x = -\frac{1}{6}(y - 15)^2 - \frac{5}{2}\)[/tex]
Thus, the correct answer is option D.
