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Find the focus and directrix of the following parabola:

[tex]\[ (y-4)^2=16(x-6) \][/tex]

Focus: [tex]\((\square, \square)\)[/tex]

Directrix: [tex]\(x = \square\)[/tex]


Sagot :

To find the focus and directrix of the given parabola, we'd look at its standard form and parameters. The given equation is:

[tex]\[ (y - 4)^2 = 16(x - 6) \][/tex]

This equation is in the form [tex]\((y - k)^2 = 4p(x - h)\)[/tex], with [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex] being parameters of the parabola.

1. Identify the vertex coordinates (h, k):
By comparing [tex]\((y - 4)^2 = 16(x - 6)\)[/tex] with [tex]\((y - k)^2 = 4p(x - h)\)[/tex], we see that:
- [tex]\(k = 4\)[/tex]
- [tex]\(h = 6\)[/tex]

Therefore, the vertex of the parabola is at [tex]\((6, 4)\)[/tex].

2. Determine the value of [tex]\(p\)[/tex]:
From the standard form, [tex]\(4p\)[/tex] is the coefficient of [tex]\((x-h)\)[/tex]. Here, [tex]\(4p = 16\)[/tex]. Solving for [tex]\(p\)[/tex]:

[tex]\[ p = \frac{16}{4} = 4 \][/tex]

3. Find the coordinates of 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]. With [tex]\(h = 6\)[/tex], [tex]\(k = 4\)[/tex], and [tex]\(p = 4\)[/tex]:

[tex]\[ \text{Focus: } (h + p, k) = (6 + 4, 4) = (10, 4) \][/tex]

4. 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]. With [tex]\(h = 6\)[/tex] and [tex]\(p = 4\)[/tex]:

[tex]\[ \text{Directrix: } x = h - p = 6 - 4 = 2 \][/tex]

Thus, the focus and the directrix of the parabola are:

[tex]\[ \text{Focus: } (10, 4) \][/tex]
[tex]\[ \text{Directrix: } x = 2 \][/tex]