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Sagot :
Let's determine which equation represents the parabola with a focus at [tex]\((0,0)\)[/tex] and a directrix of [tex]\(y = 2\)[/tex].
### Step-by-Step Solution:
1. Identify the Standard Form of the Parabola:
The standard form of a parabola with a vertical axis of symmetry and vertex at [tex]\((h, k)\)[/tex] is given by:
[tex]\[ (x - h)^2 = 4p(y - k) \][/tex]
where:
- [tex]\((h, k)\)[/tex] is the vertex
- [tex]\(p\)[/tex] is the distance from the vertex to the focus
2. Given Information:
- Focus: [tex]\((0,0)\)[/tex]
- Directrix: [tex]\(y = 2\)[/tex]
3. Determine the Vertex:
The vertex of the parabola is halfway between the focus and the directrix.
- The focus [tex]\((0,0)\)[/tex] is at [tex]\(y = 0\)[/tex].
- The directrix is at [tex]\(y = 2\)[/tex].
The midpoint (vertex) of the line segment joining these two points is:
[tex]\[ k = \frac{0 + 2}{2} = 1 \][/tex]
So, the vertex is [tex]\((0, 1)\)[/tex]. Therefore, [tex]\(h = 0\)[/tex] and [tex]\(k = 1\)[/tex].
4. Determine the Value of [tex]\(p\)[/tex]:
The distance [tex]\(p\)[/tex] is the distance from the vertex to the focus, which is:
[tex]\[ p = 1 - 0 = 1 \][/tex]
Since the parabola opens downwards (focus is below the vertex), [tex]\(p\)[/tex] should be negative:
[tex]\[ p = -1 \][/tex]
5. Substitute the Values into the Standard Form Equation:
Given that [tex]\(h = 0\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(p = -1\)[/tex], we substitute into the standard form equation:
[tex]\[ (x - 0)^2 = 4(-1)(y - 1) \][/tex]
Simplifying this, we get:
[tex]\[ x^2 = -4(y - 1) \][/tex]
6. Compare with the Given Options:
Now, we compare this equation with the given options:
- [tex]\(x^2 = -(y - 1)\)[/tex]
- [tex]\(x^2 = -4(y - 1)\)[/tex]
- [tex]\(x^2 = -4y\)[/tex]
- [tex]\(x^2 = -y\)[/tex]
Clearly, the correct equation that matches our derived equation is:
[tex]\[ x^2 = -4(y - 1) \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{x^2 = -4(y - 1)} \][/tex]
### Step-by-Step Solution:
1. Identify the Standard Form of the Parabola:
The standard form of a parabola with a vertical axis of symmetry and vertex at [tex]\((h, k)\)[/tex] is given by:
[tex]\[ (x - h)^2 = 4p(y - k) \][/tex]
where:
- [tex]\((h, k)\)[/tex] is the vertex
- [tex]\(p\)[/tex] is the distance from the vertex to the focus
2. Given Information:
- Focus: [tex]\((0,0)\)[/tex]
- Directrix: [tex]\(y = 2\)[/tex]
3. Determine the Vertex:
The vertex of the parabola is halfway between the focus and the directrix.
- The focus [tex]\((0,0)\)[/tex] is at [tex]\(y = 0\)[/tex].
- The directrix is at [tex]\(y = 2\)[/tex].
The midpoint (vertex) of the line segment joining these two points is:
[tex]\[ k = \frac{0 + 2}{2} = 1 \][/tex]
So, the vertex is [tex]\((0, 1)\)[/tex]. Therefore, [tex]\(h = 0\)[/tex] and [tex]\(k = 1\)[/tex].
4. Determine the Value of [tex]\(p\)[/tex]:
The distance [tex]\(p\)[/tex] is the distance from the vertex to the focus, which is:
[tex]\[ p = 1 - 0 = 1 \][/tex]
Since the parabola opens downwards (focus is below the vertex), [tex]\(p\)[/tex] should be negative:
[tex]\[ p = -1 \][/tex]
5. Substitute the Values into the Standard Form Equation:
Given that [tex]\(h = 0\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(p = -1\)[/tex], we substitute into the standard form equation:
[tex]\[ (x - 0)^2 = 4(-1)(y - 1) \][/tex]
Simplifying this, we get:
[tex]\[ x^2 = -4(y - 1) \][/tex]
6. Compare with the Given Options:
Now, we compare this equation with the given options:
- [tex]\(x^2 = -(y - 1)\)[/tex]
- [tex]\(x^2 = -4(y - 1)\)[/tex]
- [tex]\(x^2 = -4y\)[/tex]
- [tex]\(x^2 = -y\)[/tex]
Clearly, the correct equation that matches our derived equation is:
[tex]\[ x^2 = -4(y - 1) \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{x^2 = -4(y - 1)} \][/tex]
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