To determine the coordinates of the focus of the given parabola, we'll first convert the equation into the standard form of a parabola and then use the properties of a parabola to find the focus.
The given equation is:
[tex]\[
\frac{1}{32}(y-2)^2 = x-1
\][/tex]
Let's rewrite this equation in a more familiar form. Multiply both sides by 32 to eliminate the fraction:
[tex]\[
(y-2)^2 = 32(x-1)
\][/tex]
We now have the equation in a form resembling the standard equation of a parabola that opens horizontally:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
Comparing [tex]\((y - 2)^2 = 32(x - 1)\)[/tex] with the standard form [tex]\((y - k)^2 = 4p(x - h)\)[/tex]:
- [tex]\(h = 1\)[/tex]
- [tex]\(k = 2\)[/tex]
- [tex]\(4p = 32\)[/tex]
From this, we solve for [tex]\(p\)[/tex]:
[tex]\[
4p = 32 \implies p = 8
\][/tex]
The vertex of the parabola is at [tex]\((h, k) = (1, 2)\)[/tex]. For a parabola that opens horizontally, the focus is located at:
[tex]\((h + p, k)\)[/tex]
Substituting [tex]\(h = 1\)[/tex], [tex]\(k = 2\)[/tex], and [tex]\(p = 8\)[/tex]:
[tex]\[
\text{Focus} = (1 + 8, 2) = (9, 2)
\][/tex]
Therefore, the coordinates of the focus are [tex]\((9, 2)\)[/tex].
Out of the given options, the correct one is:
[tex]\((9, 2)\)[/tex]
