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Sagot :
To determine which circles lie entirely within the fourth quadrant, let's examine the given equations of the circles in standard form:
1. Circle A: [tex]\((x-5)^2 + (y+7)^2 = 16\)[/tex]
- Center: [tex]\((5, -7)\)[/tex]
- Radius: [tex]\(\sqrt{16} = 4\)[/tex]
2. Circle B: [tex]\((x+3)^2 + (y-2)^2 = 25\)[/tex]
- Center: [tex]\((-3, 2)\)[/tex]
- Radius: [tex]\(\sqrt{25} = 5\)[/tex]
3. Circle C: [tex]\((x-3)^2 + (y+4)^2 = 1\)[/tex]
- Center: [tex]\((3, -4)\)[/tex]
- Radius: [tex]\(\sqrt{1} = 1\)[/tex]
4. Circle D: [tex]\((x-4)^2 + (y+2)^2 = 32\)[/tex]
- Center: [tex]\((4, -2)\)[/tex]
- Radius: [tex]\(\sqrt{32} \approx 5.66\)[/tex]
The fourth quadrant is defined where [tex]\(x > 0\)[/tex] and [tex]\(y < 0\)[/tex]. Also, for a circle to lie completely within the fourth quadrant, its top part should not cross beyond the x-axis (i.e., [tex]\(y + \text{radius} < 0\)[/tex]) and its left part should not cross beyond the y-axis (i.e., [tex]\(x - \text{radius} > 0\)[/tex]).
Now, we evaluate each circle:
### Circle A:
- Center: [tex]\((5, -7)\)[/tex]
- Radius: [tex]\(4\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(5 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-7 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y\ + \text{radius} = -7 + 4 = -3 > 0\)[/tex] (fails the [tex]\(y + \text{radius} < 0\)[/tex] check)
Circle A does not lie completely within the fourth quadrant.
### Circle B:
- Center: [tex]\((-3, 2)\)[/tex]
- Radius: [tex]\(5\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(-3 < 0\)[/tex] (fails [tex]\(x > 0\)[/tex] check)
- [tex]\(y\)[/tex]-coordinate: [tex]\(2 > 0\)[/tex] (fails [tex]\(y < 0\)[/tex] check)
Circle B does not lie completely within the fourth quadrant.
### Circle C:
- Center: [tex]\((3, -4)\)[/tex]
- Radius: [tex]\(1\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(3 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-4 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y + \text{radius} = -4 + 1 = -3 < 0\)[/tex] (satisfies [tex]\(y + \text{radius} < 0\)[/tex] check)
- [tex]\(x - \text{radius} = 3 - 1 = 2 > 0\)[/tex] (satisfies [tex]\(x - \text{radius} > 0\)[/tex] check)
Circle C lies completely within the fourth quadrant.
### Circle D:
- Center: [tex]\((4, -2)\)[/tex]
- Radius: [tex]\(\sqrt{32} \approx 5.66\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(4 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-2 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y + \text{radius} = -2 + 5.66 > 0\)[/tex] (fails [tex]\(y + \text{radius} < 0\)[/tex] check)
Circle D does not lie completely within the fourth quadrant.
Summary:
The circle that lies completely within the fourth quadrant is:
- Circle C: [tex]\((x-3)^2 + (y+4)^2 = 1\)[/tex]
Therefore, the correct answer is only Circle C, hence we submit "C."
1. Circle A: [tex]\((x-5)^2 + (y+7)^2 = 16\)[/tex]
- Center: [tex]\((5, -7)\)[/tex]
- Radius: [tex]\(\sqrt{16} = 4\)[/tex]
2. Circle B: [tex]\((x+3)^2 + (y-2)^2 = 25\)[/tex]
- Center: [tex]\((-3, 2)\)[/tex]
- Radius: [tex]\(\sqrt{25} = 5\)[/tex]
3. Circle C: [tex]\((x-3)^2 + (y+4)^2 = 1\)[/tex]
- Center: [tex]\((3, -4)\)[/tex]
- Radius: [tex]\(\sqrt{1} = 1\)[/tex]
4. Circle D: [tex]\((x-4)^2 + (y+2)^2 = 32\)[/tex]
- Center: [tex]\((4, -2)\)[/tex]
- Radius: [tex]\(\sqrt{32} \approx 5.66\)[/tex]
The fourth quadrant is defined where [tex]\(x > 0\)[/tex] and [tex]\(y < 0\)[/tex]. Also, for a circle to lie completely within the fourth quadrant, its top part should not cross beyond the x-axis (i.e., [tex]\(y + \text{radius} < 0\)[/tex]) and its left part should not cross beyond the y-axis (i.e., [tex]\(x - \text{radius} > 0\)[/tex]).
Now, we evaluate each circle:
### Circle A:
- Center: [tex]\((5, -7)\)[/tex]
- Radius: [tex]\(4\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(5 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-7 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y\ + \text{radius} = -7 + 4 = -3 > 0\)[/tex] (fails the [tex]\(y + \text{radius} < 0\)[/tex] check)
Circle A does not lie completely within the fourth quadrant.
### Circle B:
- Center: [tex]\((-3, 2)\)[/tex]
- Radius: [tex]\(5\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(-3 < 0\)[/tex] (fails [tex]\(x > 0\)[/tex] check)
- [tex]\(y\)[/tex]-coordinate: [tex]\(2 > 0\)[/tex] (fails [tex]\(y < 0\)[/tex] check)
Circle B does not lie completely within the fourth quadrant.
### Circle C:
- Center: [tex]\((3, -4)\)[/tex]
- Radius: [tex]\(1\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(3 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-4 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y + \text{radius} = -4 + 1 = -3 < 0\)[/tex] (satisfies [tex]\(y + \text{radius} < 0\)[/tex] check)
- [tex]\(x - \text{radius} = 3 - 1 = 2 > 0\)[/tex] (satisfies [tex]\(x - \text{radius} > 0\)[/tex] check)
Circle C lies completely within the fourth quadrant.
### Circle D:
- Center: [tex]\((4, -2)\)[/tex]
- Radius: [tex]\(\sqrt{32} \approx 5.66\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(4 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-2 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y + \text{radius} = -2 + 5.66 > 0\)[/tex] (fails [tex]\(y + \text{radius} < 0\)[/tex] check)
Circle D does not lie completely within the fourth quadrant.
Summary:
The circle that lies completely within the fourth quadrant is:
- Circle C: [tex]\((x-3)^2 + (y+4)^2 = 1\)[/tex]
Therefore, the correct answer is only Circle C, hence we submit "C."
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