To find the focus and directrix of the given parabola [tex]\((y - 4)^2 = 16(x - 6)\)[/tex], we'll start by identifying key features from the equation. The standard form for a horizontal parabola is [tex]\((y - k)^2 = 4p(x - h)\)[/tex], where:
- [tex]\((h, k)\)[/tex] represents the vertex of the parabola.
- [tex]\(p\)[/tex] is the distance from the vertex to the focus (and also to the directrix).
1. Identify the vertex:
Comparing [tex]\((y - 4)^2 = 16(x - 6)\)[/tex] with the standard form [tex]\((y - k)^2 = 4p(x - h)\)[/tex]:
- [tex]\(h = 6\)[/tex]
- [tex]\(k = 4\)[/tex]
Hence, the vertex of the parabola is [tex]\((6, 4)\)[/tex].
2. Determine [tex]\(p\)[/tex]:
From the equation [tex]\((y - 4)^2 = 16(x - 6)\)[/tex], we can see that [tex]\(4p = 16\)[/tex]. Solving for [tex]\(p\)[/tex]:
[tex]\[
4p = 16 \implies p = 4
\][/tex]
3. Find the focus:
Since this is a horizontal parabola (opening to the right), the focus will be [tex]\(p\)[/tex] units to the right of the vertex:
- Vertex: [tex]\((6, 4)\)[/tex]
- Focus: [tex]\((6 + 4, 4) = (10, 4)\)[/tex]
4. Find the directrix:
The directrix of a horizontal parabola is a vertical line [tex]\(p\)[/tex] units to the left of the vertex:
- Vertex: [tex]\((6, 4)\)[/tex]
- Directrix: [tex]\(x = 6 - 4 = 2\)[/tex]
Therefore, the focus and directrix of the parabola [tex]\((y - 4)^2 = 16(x - 6)\)[/tex] are:
- Focus: [tex]\((10, 4)\)[/tex]
- Directrix: [tex]\(x = 2\)[/tex]
