To find the focus and directrix of the given parabola [tex]\((x+5)^2=16(y-3)\)[/tex], let's follow the steps below:
### Step-by-Step Solution
1. Identify the standard form of the parabola:
The given equation of the parabola is:
[tex]\[
(x + 5)^2 = 16(y - 3)
\][/tex]
We can compare this with the standard form of a vertical parabola [tex]\((x - h)^2 = 4p(y - k)\)[/tex].
2. Determine the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex]:
- From [tex]\( (x + 5)^2 \)[/tex], we can see that [tex]\((x - (-5))^2\)[/tex]. Hence, [tex]\(h = -5\)[/tex].
- From [tex]\(16(y - 3)\)[/tex], we observe that [tex]\(16\)[/tex] can be written as [tex]\(4 \times 4\)[/tex]. Therefore, [tex]\(4p = 16\)[/tex] which implies [tex]\(p = 4\)[/tex].
- The term [tex]\((y - 3)\)[/tex] corresponds directly with [tex]\((y - k)\)[/tex]. Thus, [tex]\(k = 3\)[/tex].
3. Find the focus of the parabola:
The focus of a parabola given by [tex]\((x - h)^2 = 4p(y - k)\)[/tex] is located at [tex]\((h, k + p)\)[/tex].
- We have [tex]\(h = -5\)[/tex], [tex]\(k = 3\)[/tex], and [tex]\(p = 4\)[/tex].
- Therefore, the focus is:
[tex]\[
(h, k + p) = (-5, 3 + 4) = (-5, 7)
\][/tex]
4. Find the directrix of the parabola:
The directrix of a parabola given by [tex]\((x - h)^2 = 4p(y - k)\)[/tex] is the line [tex]\(y = k - p\)[/tex].
- We have [tex]\(k = 3\)[/tex] and [tex]\(p = 4\)[/tex].
- Therefore, the directrix is:
[tex]\[
y = k - p = 3 - 4 = -1
\][/tex]
### Conclusion
- The focus of the parabola is [tex]\((-5, 7)\)[/tex].
- The directrix of the parabola is [tex]\(y = -1\)[/tex].
So, we can now fill in the blanks:
Focus: ( [tex]\(-5\)[/tex], [tex]\(7\)[/tex] )
Directrix: [tex]\( y = -1 \)[/tex]
