Find the focus and directrix of the following parabola:

[tex]\[
(x-5)^2 = 8(y-6)
\][/tex]

Focus: ([?], [?])

Directrix: [tex]\( y = [?] \)[/tex]



Answer :

To find the focus and directrix of the given parabola, let’s start by transforming and analyzing the given equation:

Given parabola:
[tex]\[ (x-5)^2 = 8(y-6) \][/tex]

This equation is in the form [tex]\( (x-h)^2 = 4p(y-k) \)[/tex], which represents a parabola with a vertical axis. By comparing the given equation with the standard form, we identify the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(4p\)[/tex]:

- [tex]\(h = 5\)[/tex]
- [tex]\(k = 6\)[/tex]
- [tex]\(4p = 8\)[/tex]

To find [tex]\(p\)[/tex]:

[tex]\[ p = \frac{4p}{4} = \frac{8}{4} = 2.0 \][/tex]

Now we have identified the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex]:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = 6\)[/tex]
- [tex]\(p = 2.0\)[/tex]

The focus of the parabola is found at [tex]\((h, k + p)\)[/tex]:
[tex]\[ \text{Focus} = (5, 6 + 2.0) = (5, 8.0) \][/tex]

The directrix of the parabola is given by the line [tex]\( y = k - p \)[/tex]:
[tex]\[ \text{Directrix} = y = 6 - 2.0 = 4.0 \][/tex]

Therefore, the focus and directrix of the given parabola are:

[tex]\[ \text{Focus: } (5, 8.0) \][/tex]

[tex]\[ \text{Directrix: } y = 4.0 \][/tex]