To determine the equation of the directrix for the given parabola [tex]\((y - 3)^2 = 8(x - 5)\)[/tex], we will follow these steps:
1. Identify the standard form:
The equation [tex]\((y - 3)^2 = 8(x - 5)\)[/tex] is in the form [tex]\((y - k)^2 = 4p(x - h)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola and [tex]\(p\)[/tex] is the distance from the vertex to the focus (and also from the vertex to the directrix).
2. Determine the parameters:
From the equation [tex]\((y - 3)^2 = 8(x - 5)\)[/tex], we compare it to the standard form [tex]\((y - k)^2 = 4p(x - h)\)[/tex]:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = 3\)[/tex]
- [tex]\(4p = 8\)[/tex], which gives [tex]\(p = 2\)[/tex]
3. Find the directrix:
For a parabola in the form [tex]\((y - k)^2 = 4p(x - h)\)[/tex], the directrix is given by the line [tex]\(x = h - p\)[/tex]:
- Here, [tex]\(h = 5\)[/tex] and [tex]\(p = 2\)[/tex]
- Thus, the directrix is [tex]\(x = 5 - 2\)[/tex]
4. Calculate the directrix equation:
- [tex]\(x = 3\)[/tex]
Therefore, the equation of the directrix is [tex]\(\boxed{x = 3}\)[/tex].
Thus, the correct answer is B. [tex]\( x = 3 \)[/tex].
