To find the focus and directrix of the given parabola
[tex]\[ (y + 3)^2 = 16(x - 4), \][/tex]
we can follow these steps:
### Step 1: Identify the Standard Form
The equation can be written in the form
[tex]\[ (y - k)^2 = 4p(x - h), \][/tex]
which describes a horizontal parabola opening to the right.
Comparing with the given equation [tex]\((y + 3)^2 = 16(x - 4)\)[/tex], we get:
- [tex]\(h = 4\)[/tex]
- [tex]\(k = -3\)[/tex]
- [tex]\(4p = 16\)[/tex], thus [tex]\(p = 4\)[/tex]
### Step 2: Determine the Focus
The focus of a parabola given by [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is located at [tex]\((h + p, k)\)[/tex].
So, substituting [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex]:
[tex]\[ h = 4, \quad k = -3, \quad p = 4 \][/tex]
[tex]\[ \text{Focus} = (h + p, k) = (4 + 4, -3) = (8, -3) \][/tex]
### Step 3: Determine the Directrix
The directrix of a parabola given by [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is the line [tex]\(x = h - p\)[/tex].
So, substituting [tex]\(h\)[/tex] and [tex]\(p\)[/tex]:
[tex]\[ h = 4, \quad p = 4 \][/tex]
[tex]\[ \text{Directrix} = x = h - p = 4 - 4 = 0 \][/tex]
### Conclusion
The focus and directrix of the given parabola [tex]\((y + 3)^2 = 16(x - 4)\)[/tex] are:
- Focus: [tex]\((8, -3)\)[/tex]
- Directrix: [tex]\(x = 0\)[/tex]
Therefore:
- Focus: ([>8<], [-3])
- Directrix: [tex]\(x = \)[/tex] [>0<]
