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Which of the following circles lie completely within the fourth quadrant? Check all that apply.

A. [tex]\((x-5)^2+(y+5)^2=9\)[/tex]

B. [tex]\((x-2)^2+(y+7)^2=64\)[/tex]

C. [tex]\((x-12)^2+(y+0)^2=72\)[/tex]

D. [tex]\((x-9)^2+(y+9)^2=16\)[/tex]

Sagot :

GinnyAnsweridn
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]
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