To find the coordinates of the focus of the parabola given its equation [tex]\(\frac{1}{8}(y-5)^2 = x+3\)[/tex], we need to follow these steps:
1. Convert the given equation to standard form: The standard form of a parabola that opens sideways (either to the right or the left) is given by [tex]\((y - k)^2 = 4p(x - h)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex and [tex]\(p\)[/tex] is the distance from the vertex to the focus.
2. Rewrite the given equation: Start with the given equation:
[tex]\[
\frac{1}{8}(y-5)^2 = x + 3
\][/tex]
Multiply both sides of the equation by 8 to get rid of the fraction:
[tex]\[
(y - 5)^2 = 8(x + 3)
\][/tex]
3. Identify the vertex [tex]\((h, k)\)[/tex]: From the equation [tex]\((y - 5)^2 = 8(x + 3)\)[/tex], we can see that it matches the standard form [tex]\((y - k)^2 = 4p(x - h)\)[/tex] where [tex]\(k = 5\)[/tex] and [tex]\(h = -3\)[/tex].
4. Find the value of [tex]\(p\)[/tex]: The coefficient on the right side of the equation is [tex]\(8\)[/tex], which is equal to [tex]\(4p\)[/tex]. Therefore, we have:
[tex]\[
4p = 8 \implies p = 2
\][/tex]
5. Determine the coordinates of the focus: The focus of the parabola is located at [tex]\((h + p, k)\)[/tex] for a parabola that opens to the right (because [tex]\(p > 0\)[/tex]).
Substitute [tex]\(h = -3\)[/tex], [tex]\(k = 5\)[/tex], and [tex]\(p = 2\)[/tex]:
[tex]\[
\text{Focus} = (h + p, k) = (-3 + 2, 5) = (-1, 5)
\][/tex]
Thus, the coordinates of the focus are [tex]\(\boxed{(-1, 5)}\)[/tex].
