To find the focus and directrix of the given parabola equation:
[tex]\[
(y - 1)^2 = 4(x - 1)
\][/tex]
we should start by rewriting it in the standard form of a parabola that opens horizontally. The standard form is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(p\)[/tex] is the distance from the vertex to the focus (and also to the directrix).
In this equation:
- Matching [tex]\((y - 1)^2 = 4(x - 1)\)[/tex] to the standard form, we identify [tex]\(h = 1\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(4p = 4\)[/tex].
We solve for [tex]\(p\)[/tex]:
- [tex]\(p = \frac{4}{4} = 1\)[/tex]
The focus of the parabola is found using [tex]\((h + p, k)\)[/tex]:
- [tex]\(h + p = 1 + 1 = 2\)[/tex]
- [tex]\(k = 1\)[/tex]
So, the coordinates of the focus are:
[tex]\[
(2, 1)
\][/tex]
The directrix is a vertical line given by [tex]\(x = h - p\)[/tex]:
- [tex]\(h - p = 1 - 1 = 0\)[/tex]
So, the directrix is:
[tex]\[
x = 0
\][/tex]
Therefore, the focus and directrix of the parabola [tex]\((y - 1)^2 = 4(x - 1)\)[/tex] are:
Focus: [tex]\((2, 1)\)[/tex]
Directrix: [tex]\(x = 0\)[/tex]
So the final answers are:
[tex]\[
\text{Focus: } (2, 1)
\][/tex]
[tex]\[
\text{Directrix: } x = 0
\][/tex]
