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Sagot :
To determine which circles lie completely within the fourth quadrant, we need to examine the center and radius of each circle. The fourth quadrant is defined as the region where both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates are positive.
Let's analyze each circle:
### Circle A: [tex]\((x-5)^2 + (y+5)^2 = 9\)[/tex]
- Center: [tex]\((5, -5)\)[/tex]
- Radius: The radius can be computed as the square root of 9, which is 3.
For the circle to lie completely within the fourth quadrant:
- The center must be in the fourth quadrant, implying [tex]\(y\)[/tex] must be negative.
- [tex]\(x - \text{radius}\)[/tex] and [tex]\(y - \text{radius}\)[/tex] must both remain positive.
Here:
- [tex]\(x\)[/tex] coordinate: [tex]\(5\)[/tex], which is positive.
- [tex]\(y\)[/tex] coordinate: [tex]\(-5\)[/tex], which is negative. Hence, the circle is not in the fourth quadrant.
### Circle B: [tex]\((x-2)^2 + (y+7)^2 = 64\)[/tex]
- Center: [tex]\((2, -7)\)[/tex]
- Radius: The radius is the square root of 64, which is 8.
For the circle to lie completely within the fourth quadrant:
- [tex]\(x - \text{radius} > 0\)[/tex]
- [tex]\(y - \text{radius} > 0\)[/tex]
Here:
- [tex]\(x\)[/tex] coordinate: [tex]\(2\)[/tex], radius: [tex]\(8\)[/tex]. [tex]\(2 - 8 = -6\)[/tex], negative
- [tex]\(y\)[/tex] coordinate: [tex]\(-7\)[/tex], which is negative. The center is not in the fourth quadrant.
### Circle C: [tex]\((x-12)^2 + (y+0)^2 = 72\)[/tex]
- Center: [tex]\((12, 0)\)[/tex]
- Radius: The radius is the square root of 72, which is approximately 8.49.
For the circle to lie completely within the fourth quadrant:
- [tex]\(x - \text{radius} > 0\)[/tex]
- [tex]\(y - \text{radius} > 0\)[/tex]
Here:
- [tex]\(x\)[/tex] coordinate: [tex]\(12\)[/tex], radius: [tex]\(8.49\)[/tex]. [tex]\(12 - 8.49 > 0\)[/tex], positive.
- [tex]\(y\)[/tex] coordinate: [tex]\(0\)[/tex], radius [tex]\(8.49\)[/tex]. [tex]\(0 - 8.49 = -8.49\)[/tex], negative
The center is not in the fourth quadrant.
### Circle D: [tex]\((x-9)^2 + (y+9)^2 = 16\)[/tex]
- Center: [tex]\((9, -9)\)[/tex]
- Radius: The radius is the square root of 16, which is 4.
For the circle to lie completely within the fourth quadrant:
- [tex]\(x - \text{radius} > 0\)[/tex]
- [tex]\(y - \text{radius} > 0\)[/tex]
Here:
- [tex]\(x\)[/tex] coordinate: [tex]\(9\)[/tex], radius: [tex]\(4\)[/tex]. [tex]\(9 - 4 = 5\)[/tex], positive.
- [tex]\(y\)[/tex] coordinate: [tex]\(-9\)[/tex], radius: [tex]\(4\)[/tex]. [tex]\(-9 - 4 = -13\)[/tex], negative.
The center is not in the fourth quadrant.
### Conclusion
None of the given circles lie completely within the fourth quadrant.
Thus, the answer is:
[tex]\[ \boxed{[]} \][/tex]
Let's analyze each circle:
### Circle A: [tex]\((x-5)^2 + (y+5)^2 = 9\)[/tex]
- Center: [tex]\((5, -5)\)[/tex]
- Radius: The radius can be computed as the square root of 9, which is 3.
For the circle to lie completely within the fourth quadrant:
- The center must be in the fourth quadrant, implying [tex]\(y\)[/tex] must be negative.
- [tex]\(x - \text{radius}\)[/tex] and [tex]\(y - \text{radius}\)[/tex] must both remain positive.
Here:
- [tex]\(x\)[/tex] coordinate: [tex]\(5\)[/tex], which is positive.
- [tex]\(y\)[/tex] coordinate: [tex]\(-5\)[/tex], which is negative. Hence, the circle is not in the fourth quadrant.
### Circle B: [tex]\((x-2)^2 + (y+7)^2 = 64\)[/tex]
- Center: [tex]\((2, -7)\)[/tex]
- Radius: The radius is the square root of 64, which is 8.
For the circle to lie completely within the fourth quadrant:
- [tex]\(x - \text{radius} > 0\)[/tex]
- [tex]\(y - \text{radius} > 0\)[/tex]
Here:
- [tex]\(x\)[/tex] coordinate: [tex]\(2\)[/tex], radius: [tex]\(8\)[/tex]. [tex]\(2 - 8 = -6\)[/tex], negative
- [tex]\(y\)[/tex] coordinate: [tex]\(-7\)[/tex], which is negative. The center is not in the fourth quadrant.
### Circle C: [tex]\((x-12)^2 + (y+0)^2 = 72\)[/tex]
- Center: [tex]\((12, 0)\)[/tex]
- Radius: The radius is the square root of 72, which is approximately 8.49.
For the circle to lie completely within the fourth quadrant:
- [tex]\(x - \text{radius} > 0\)[/tex]
- [tex]\(y - \text{radius} > 0\)[/tex]
Here:
- [tex]\(x\)[/tex] coordinate: [tex]\(12\)[/tex], radius: [tex]\(8.49\)[/tex]. [tex]\(12 - 8.49 > 0\)[/tex], positive.
- [tex]\(y\)[/tex] coordinate: [tex]\(0\)[/tex], radius [tex]\(8.49\)[/tex]. [tex]\(0 - 8.49 = -8.49\)[/tex], negative
The center is not in the fourth quadrant.
### Circle D: [tex]\((x-9)^2 + (y+9)^2 = 16\)[/tex]
- Center: [tex]\((9, -9)\)[/tex]
- Radius: The radius is the square root of 16, which is 4.
For the circle to lie completely within the fourth quadrant:
- [tex]\(x - \text{radius} > 0\)[/tex]
- [tex]\(y - \text{radius} > 0\)[/tex]
Here:
- [tex]\(x\)[/tex] coordinate: [tex]\(9\)[/tex], radius: [tex]\(4\)[/tex]. [tex]\(9 - 4 = 5\)[/tex], positive.
- [tex]\(y\)[/tex] coordinate: [tex]\(-9\)[/tex], radius: [tex]\(4\)[/tex]. [tex]\(-9 - 4 = -13\)[/tex], negative.
The center is not in the fourth quadrant.
### Conclusion
None of the given circles lie completely within the fourth quadrant.
Thus, the answer is:
[tex]\[ \boxed{[]} \][/tex]
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