Answer :
Let's determine if the centers of the given circles lie in the third quadrant. The third quadrant is defined by the condition that both the \( x \) and \( y \) coordinates of the center must be negative.
For each circle, we'll analyze the information provided:
### Circle A: \((x+16)^2 + (y+3)^2 = 17\)
- Center:
[tex]\[(x+16)^2 \rightarrow x + 16 = 0 \implies x = -16\][/tex]
[tex]\[(y+3)^2 \rightarrow y + 3 = 0 \implies y = -3\][/tex]
- Coordinates of Center: \((-16, -3)\)
### Circle B: \((x+14)^2 + (y-14)^2 = 84\)
- Center:
[tex]\[(x+14)^2 \rightarrow x + 14 = 0 \implies x = -14\][/tex]
[tex]\[(y-14)^2 \rightarrow y - 14 = 0 \implies y = 14\][/tex]
- Coordinates of Center: \((-14, 14)\)
### Circle C: \((x+3)^2 + (y-6)^2 = 44\)
- Center:
[tex]\[(x+3)^2 \rightarrow x + 3 = 0 \implies x = -3\][/tex]
[tex]\[(y-6)^2 \rightarrow y - 6 = 0 \implies y = 6\][/tex]
- Coordinates of Center: \((-3, 6)\)
### Circle D: \((x+9)^2 + (y+12)^2 = 36\)
- Center:
[tex]\[(x+9)^2 \rightarrow x + 9 = 0 \implies x = -9\][/tex]
[tex]\[(y+12)^2 \rightarrow y + 12 = 0 \implies y = -12\][/tex]
- Coordinates of Center: \((-9, -12)\)
#### Analysis:
- The center of Circle A is \((-16, -3)\), which is in the third quadrant because both coordinates are negative.
- The center of Circle B is \((-14, 14)\), which is not in the third quadrant since the \( y \)-coordinate is positive.
- The center of Circle C is \((-3, 6)\), which is not in the third quadrant since the \( y \)-coordinate is positive.
- The center of Circle D is \((-9, -12)\), which is in the third quadrant because both coordinates are negative.
### Conclusion:
Circles with their centers in the third quadrant are:
- Circle A
- Circle D
So, the circles that have their centers in the third quadrant are A and D.
For each circle, we'll analyze the information provided:
### Circle A: \((x+16)^2 + (y+3)^2 = 17\)
- Center:
[tex]\[(x+16)^2 \rightarrow x + 16 = 0 \implies x = -16\][/tex]
[tex]\[(y+3)^2 \rightarrow y + 3 = 0 \implies y = -3\][/tex]
- Coordinates of Center: \((-16, -3)\)
### Circle B: \((x+14)^2 + (y-14)^2 = 84\)
- Center:
[tex]\[(x+14)^2 \rightarrow x + 14 = 0 \implies x = -14\][/tex]
[tex]\[(y-14)^2 \rightarrow y - 14 = 0 \implies y = 14\][/tex]
- Coordinates of Center: \((-14, 14)\)
### Circle C: \((x+3)^2 + (y-6)^2 = 44\)
- Center:
[tex]\[(x+3)^2 \rightarrow x + 3 = 0 \implies x = -3\][/tex]
[tex]\[(y-6)^2 \rightarrow y - 6 = 0 \implies y = 6\][/tex]
- Coordinates of Center: \((-3, 6)\)
### Circle D: \((x+9)^2 + (y+12)^2 = 36\)
- Center:
[tex]\[(x+9)^2 \rightarrow x + 9 = 0 \implies x = -9\][/tex]
[tex]\[(y+12)^2 \rightarrow y + 12 = 0 \implies y = -12\][/tex]
- Coordinates of Center: \((-9, -12)\)
#### Analysis:
- The center of Circle A is \((-16, -3)\), which is in the third quadrant because both coordinates are negative.
- The center of Circle B is \((-14, 14)\), which is not in the third quadrant since the \( y \)-coordinate is positive.
- The center of Circle C is \((-3, 6)\), which is not in the third quadrant since the \( y \)-coordinate is positive.
- The center of Circle D is \((-9, -12)\), which is in the third quadrant because both coordinates are negative.
### Conclusion:
Circles with their centers in the third quadrant are:
- Circle A
- Circle D
So, the circles that have their centers in the third quadrant are A and D.
