Sure! To find the focus and the directrix of the parabola given by the equation [tex]\((x - 4)^2 = 16(y + 2)\)[/tex], we will follow these steps:
1. Identify the form of the equation:
The given equation is [tex]\((x - 4)^2 = 16(y + 2)\)[/tex].
2. Rewrite the equation to standard form:
The standard form of a parabola that opens vertically is [tex]\((x - h)^2 = 4p(y - k)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex, and [tex]\(p\)[/tex] is the distance from the vertex to the focus and the directrix.
3. Extract the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex]:
By comparing [tex]\((x - 4)^2 = 16(y + 2)\)[/tex] to [tex]\((x - h)^2 = 4p(y - k)\)[/tex],
- [tex]\(h = 4\)[/tex],
- [tex]\(k = -2\)[/tex],
- [tex]\(4p = 16\)[/tex].
Solving for [tex]\(p\)[/tex], we get [tex]\(p = \frac{16}{4} = 4\)[/tex].
4. Determine the focus:
The focus of a parabola that opens vertically is located at [tex]\((h, k + p)\)[/tex].
- Substituting [tex]\(h = 4\)[/tex], [tex]\(k = -2\)[/tex], and [tex]\(p = 4\)[/tex],
the coordinates of the focus are [tex]\((4, -2 + 4) = (4, 2)\)[/tex].
5. Determine the directrix:
The directrix of a parabola that opens vertically is given by the line [tex]\(y = k - p\)[/tex].
- Substituting [tex]\(k = -2\)[/tex] and [tex]\(p = 4\)[/tex],
the equation of the directrix is [tex]\(y = -2 - 4 = -6\)[/tex].
Thus, the focus of the parabola [tex]\((x - 4)^2 = 16(y + 2)\)[/tex] is [tex]\((4, 2)\)[/tex], and the directrix is [tex]\(y = -6\)[/tex].
Final answers:
- Focus: [tex]\((4, 2)\)[/tex]
- Directrix: [tex]\(y = -6\)[/tex]
