To find the focus and directrix of the parabola given by the equation [tex]\((x-5)^2 = 8(y-1)\)[/tex], follow these steps:
1. Identify the standard form:
The standard form of a parabola with a vertical axis is [tex]\((x-h)^2 = 4p(y-k)\)[/tex], where [tex]\((h, k)\)[/tex] represents the vertex of the parabola.
2. Extract the vertex:
Compare the given equation [tex]\((x-5)^2 = 8(y-1)\)[/tex] to the standard form [tex]\((x-h)^2 = 4p(y-k)\)[/tex]. Here, [tex]\(h\)[/tex] is 5 and [tex]\(k\)[/tex] is 1. Therefore, the vertex of the parabola is [tex]\((5, 1)\)[/tex].
3. Find the value of [tex]\(p\)[/tex]:
By comparing [tex]\((x-5)^2 = 8(y-1)\)[/tex] with [tex]\((x-h)^2 = 4p(y-k)\)[/tex], we can identify that [tex]\(4p\)[/tex] = 8. Solving for [tex]\(p\)[/tex], we get:
[tex]\[
4p = 8 \implies p = 2
\][/tex]
4. Determine the focus:
The focus of the parabola with a vertical axis can be found using the coordinates [tex]\((h, k + p)\)[/tex]. Since [tex]\(h = 5\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(p = 2\)[/tex]:
[tex]\[
\text{Focus} = (5, 1 + 2) = (5, 3)
\][/tex]
5. Determine the directrix:
The equation of the directrix for a parabola with a vertical axis is given by [tex]\(y = k - p\)[/tex]. Substituting [tex]\(k = 1\)[/tex] and [tex]\(p = 2\)[/tex]:
[tex]\[
\text{Directrix} = y = 1 - 2 = -1
\][/tex]
Thus, the focus and directrix of the given parabola [tex]\((x-5)^2 = 8(y-1)\)[/tex] are:
Focus: [tex]\((5, 3)\)[/tex]
Directrix: [tex]\(y = -1\)[/tex]
