Answer :
To determine which circles lie completely within the fourth quadrant, we need to check the coordinates of their centers and radii. The fourth quadrant is defined by positive \(x\)-coordinates (\(x > 0\)) and negative \(y\)-coordinates (\(y < 0\)).
### Circle A: \((x-3)^2+(y+4)^2=1\)
- Center: \((3, -4)\)
- Radius: \(\sqrt{1} = 1\)
The center of Circle A is \((3, -4)\), which is located in the fourth quadrant. However, we need to check if the entire circle lies within this quadrant. For the circle to lie completely within the fourth quadrant, the entire circle should not extend into any other quadrant.
Let's look at the y-coordinate:
- The y-coordinate of the center is \( -4 \).
- The circle extends from \( y = -4 \) to \( y = -4 \pm 1 \). This means it extends from \( y = -5 \) to \( y = -3 \).
Since \(-5\) and \(-3\) are both negative, the entire circle lies within the fourth quadrant.
### Circle B: \((x-4)^2+(y+2)^2=32\)
- Center: \((4, -2)\)
- Radius: \(\sqrt{32} \approx 5.66\)
The center of Circle B is \((4, -2)\), which is located in the fourth quadrant. We need to ensure the entire radius lies within the fourth quadrant.
Let's look at the y-coordinate:
- The y-coordinate of the center is \( -2 \).
- The circle extends from \( y = -2 \) to \( y = -2 \pm \sqrt{32} \). This means it extends from \( y = -2 - 5.66 \approx -7.66 \) to \( y = -2 + 5.66 \approx 3.66 \).
Since this circle extends into the first quadrant (as it goes up to \( y = 3.66 \)), it does not lie completely within the fourth quadrant.
### Circle C: \((x-5)^2+(y+7)^2=16\)
- Center: \((5, -7)\)
- Radius: \(\sqrt{16} = 4\)
The center of Circle C is \((5, -7)\). However, from the y-coordinate of the center of Circle C (\( y = -7 \)) and checking if it expands into other quadrants:
- The circle extends from \( y = -7 \) to \( y = -7 \pm 4 \). This means it extends from \( y = -11 \) to \( y = -3 \).
Since that y range does not cross the x-axis and always remains negative, the entire circle lies within the fourth quadrant.
### Circle D: \((x+3)^2+(y-2)^2=25\)
- Center: \((-3, 2)\)
- Radius: \(\sqrt{25} = 5\)
The center of Circle D is \((-3, 2)\), which is located in the second quadrant. Since the center itself is not in the fourth quadrant, Circle D cannot lie completely within the fourth quadrant.
### Conclusion
Based on the above detailed step-by-step analysis:
- Circle A lies completely within the fourth quadrant.
- Circle B does not lie completely within the fourth quadrant.
- Circle C does not lie completely within the fourth quadrant.
- Circle D does not lie completely within the fourth quadrant.
Thus, the correct answer is:
B. [tex]\((x-4)^2+(y+2)^2=32\)[/tex].
### Circle A: \((x-3)^2+(y+4)^2=1\)
- Center: \((3, -4)\)
- Radius: \(\sqrt{1} = 1\)
The center of Circle A is \((3, -4)\), which is located in the fourth quadrant. However, we need to check if the entire circle lies within this quadrant. For the circle to lie completely within the fourth quadrant, the entire circle should not extend into any other quadrant.
Let's look at the y-coordinate:
- The y-coordinate of the center is \( -4 \).
- The circle extends from \( y = -4 \) to \( y = -4 \pm 1 \). This means it extends from \( y = -5 \) to \( y = -3 \).
Since \(-5\) and \(-3\) are both negative, the entire circle lies within the fourth quadrant.
### Circle B: \((x-4)^2+(y+2)^2=32\)
- Center: \((4, -2)\)
- Radius: \(\sqrt{32} \approx 5.66\)
The center of Circle B is \((4, -2)\), which is located in the fourth quadrant. We need to ensure the entire radius lies within the fourth quadrant.
Let's look at the y-coordinate:
- The y-coordinate of the center is \( -2 \).
- The circle extends from \( y = -2 \) to \( y = -2 \pm \sqrt{32} \). This means it extends from \( y = -2 - 5.66 \approx -7.66 \) to \( y = -2 + 5.66 \approx 3.66 \).
Since this circle extends into the first quadrant (as it goes up to \( y = 3.66 \)), it does not lie completely within the fourth quadrant.
### Circle C: \((x-5)^2+(y+7)^2=16\)
- Center: \((5, -7)\)
- Radius: \(\sqrt{16} = 4\)
The center of Circle C is \((5, -7)\). However, from the y-coordinate of the center of Circle C (\( y = -7 \)) and checking if it expands into other quadrants:
- The circle extends from \( y = -7 \) to \( y = -7 \pm 4 \). This means it extends from \( y = -11 \) to \( y = -3 \).
Since that y range does not cross the x-axis and always remains negative, the entire circle lies within the fourth quadrant.
### Circle D: \((x+3)^2+(y-2)^2=25\)
- Center: \((-3, 2)\)
- Radius: \(\sqrt{25} = 5\)
The center of Circle D is \((-3, 2)\), which is located in the second quadrant. Since the center itself is not in the fourth quadrant, Circle D cannot lie completely within the fourth quadrant.
### Conclusion
Based on the above detailed step-by-step analysis:
- Circle A lies completely within the fourth quadrant.
- Circle B does not lie completely within the fourth quadrant.
- Circle C does not lie completely within the fourth quadrant.
- Circle D does not lie completely within the fourth quadrant.
Thus, the correct answer is:
B. [tex]\((x-4)^2+(y+2)^2=32\)[/tex].
