To find the focus and the directrix of the parabola given by the equation
[tex]\[ (y - 1)^2 = 4(x - 1) \][/tex]
we can follow these steps:
1. Identify the standard form: The given equation matches the standard form of a horizontal parabola:
[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.
2. Determine the vertex: From the given equation [tex]\((y - 1)^2 = 4(x - 1)\)[/tex], we can see that:
[tex]\[
h = 1 \quad \text{and} \quad k = 1
\][/tex]
Therefore, the vertex of the parabola is [tex]\((1, 1)\)[/tex].
3. Find the value of [tex]\(p\)[/tex]: In the given equation, [tex]\(4(x - 1)\)[/tex] indicates that [tex]\(4p = 4\)[/tex]. Solving for [tex]\(p\)[/tex]:
[tex]\[
4p = 4 \implies p = 1
\][/tex]
4. Determine the coordinates of the focus: For a horizontal parabola that opens to the right (since [tex]\(p\)[/tex] is positive), the focus is located at [tex]\((h + p, k)\)[/tex]:
[tex]\[
\text{Focus} = (h + p, k) = (1 + 1, 1) = (2, 1)
\][/tex]
5. Find the equation of the directrix: The directrix of a horizontal parabola that opens to the right is a vertical line located [tex]\(p\)[/tex] units to the left of the vertex:
[tex]\[
\text{Directrix} = x = h - p = 1 - 1 = 0
\][/tex]
Summarizing, the focus and directrix of the given parabola are:
- Focus: [tex]\((2, 1)\)[/tex]
- Directrix: [tex]\(x = 0\)[/tex]
