To find the equation of the parabola, we need to determine its vertex, the value of [tex]\( p \)[/tex], and its standard form equation. Here are the steps:
1. Identifying the focus and the directrix:
- The focus is at [tex]\((9, 27)\)[/tex].
- The directrix is [tex]\( y = 11 \)[/tex].
2. Determining the vertex:
The vertex is the midpoint between the focus and the directrix. So, we calculate the vertex coordinates:
[tex]\[
\text{vertex}_x = 9
\][/tex]
[tex]\[
\text{vertex}_y = \frac{27 + 11}{2} = 19
\][/tex]
Thus, the vertex is at [tex]\((9, 19)\)[/tex].
3. Calculating [tex]\( p \)[/tex]:
[tex]\( p \)[/tex] is the distance between the vertex and the focus. This is:
[tex]\[
p = 27 - 19 = 8
\][/tex]
4. Formulating the equation of the parabola:
For a parabola opening upwards with its vertex form [tex]\((x - h)^2 = 4p(y - k)\)[/tex], and vertex [tex]\((h, k) = (9, 19)\)[/tex], we plug in the values:
[tex]\[
(x - 9)^2 = 4 \cdot 8 \cdot (y - 19)
\][/tex]
[tex]\[
(x - 9)^2 = 32(y - 19)
\][/tex]
Solving for [tex]\( y \)[/tex], we get:
[tex]\[
y - 19 = \frac{1}{32}(x - 9)^2
\][/tex]
[tex]\[
y = \frac{1}{32}(x - 9)^2 + 19
\][/tex]
Verification with provided choices:
- Compare our derived equation [tex]\( y = \frac{1}{32}(x - 9)^2 + 19 \)[/tex] with the choices provided.
None of the choices directly match this derived equation:
- A. [tex]\( f(x) = \frac{1}{32}(x + 9)^2 - 19 \)[/tex]
- B. [tex]\( f(x) = \frac{1}{16}(x + 9)^2 - 19 \)[/tex]
- C. [tex]\( f(x) = \frac{1}{16}(x - 9)^2 + 19 \)[/tex]
- D. [tex]\( f(x) = \frac{1}{32}(x - 9)^2 + 19 \)[/tex]
Given the correct match with [tex]\( y = \frac{1}{32}(x-9)^2 + 19 \)[/tex], the correct answer is:
[tex]\[ \boxed{D} \][/tex] [tex]\(f(x) = \frac{1}{32}(x - 9)^2 + 19\)[/tex].
