To find the focus and directrix of the given parabola [tex]\((x - 4)^2 = 16(y + 2)\)[/tex], we need to follow these steps:
1. Identify the form of the parabola:
The given equation [tex]\((x - 4)^2 = 16(y + 2)\)[/tex] is similar to the standard form [tex]\((x - h)^2 = 4a(y - k)\)[/tex] of a parabola that opens vertically (either upwards or downwards).
2. Extract values from the equation:
From the equation [tex]\((x - 4)^2 = 16(y + 2)\)[/tex]:
- The coordinates of the vertex [tex]\((h, k)\)[/tex] are found by comparing terms. Here, [tex]\(h = 4\)[/tex] and [tex]\(k = -2\)[/tex].
- The term [tex]\(16(y + 2)\)[/tex] can be compared with [tex]\(4a(y - k)\)[/tex] to find [tex]\(a\)[/tex]. We see that [tex]\(4a = 16\)[/tex], therefore, [tex]\(a = 4\)[/tex].
3. Determine the focus:
The formula for the focus of a parabola in the form [tex]\((x - h)^2 = 4a(y - k)\)[/tex] is [tex]\((h, k + a)\)[/tex].
- Plugging in the values: [tex]\(h = 4\)[/tex], [tex]\(k = -2\)[/tex], and [tex]\(a = 4\)[/tex], we calculate the focus as follows:
[tex]\[
\text{Focus} = (4, -2 + 4) = (4, 2)
\][/tex]
4. Determine the directrix:
The formula for the directrix of such a parabola is [tex]\(y = k - a\)[/tex].
- Using the values: [tex]\(k = -2\)[/tex] and [tex]\(a = 4\)[/tex], we calculate the directrix as follows:
[tex]\[
\text{Directrix} = y = -2 - 4 = -6
\][/tex]
So, the focus of the parabola [tex]\((x - 4)^2 = 16(y + 2)\)[/tex] is [tex]\((4, 2)\)[/tex] and the directrix is the line [tex]\(y = -6\)[/tex].
