To determine which of the given circles lie completely within the third quadrant, we need to analyze the centers and radii of each circle. The third quadrant is defined by [tex]\( x < 0 \)[/tex] and [tex]\( y < 0 \)[/tex].
### Circle A: [tex]\((x+12)^2 + (y+9)^2 = 9\)[/tex]
- Center: The equation [tex]\((x+12)^2 + (y+9)^2 = 9\)[/tex] represents a circle with center [tex]\((-12, -9)\)[/tex].
- Radius: The radius is [tex]\(\sqrt{9} = 3\)[/tex].
For the circle to lie completely within the third quadrant:
- [tex]\( x \)[/tex]-coordinate: [tex]\( -12 - 3 \geq -100 \Rightarrow -15 \geq -100 \)[/tex] (True)
- [tex]\( y \)[/tex]-coordinate: [tex]\( -9 - 3 \geq -100 \Rightarrow -12 \geq -100 \)[/tex] (True)
Since both conditions are true, Circle A lies completely within the third quadrant.
### Circle B: [tex]\((x+7)^2 + (y+7)^2 = 4\)[/tex]
- Center: The equation [tex]\((x+7)^2 + (y+7)^2 = 4\)[/tex] represents a circle with center [tex]\((-7, -7)\)[/tex].
- Radius: The radius is [tex]\(\sqrt{4} = 2\)[/tex].
For the circle to lie completely within the third quadrant:
- [tex]\( x \)[/tex]-coordinate: [tex]\( -7 - 2 \geq -100 \Rightarrow -9 \geq -100 \)[/tex] (True)
- [tex]\( y \)[/tex]-coordinate: [tex]\( -7 - 2 \geq -100 \Rightarrow -9 \geq -100 \)[/tex] (True)
Since both conditions are true, Circle B lies completely within the third quadrant.
### Circle C: [tex]\((x+5)^2 + (y+0)^2 = 7\)[/tex]
- Center: The equation [tex]\((x+5)^2 + (y+0)^2 = 7\)[/tex] represents a circle with center [tex]\((-5, 0)\)[/tex].
- Radius: The radius is [tex]\(\sqrt{7} \approx 2.64\)[/tex].
For the circle to lie completely within the third quadrant:
- [tex]\( x \)[/tex]-coordinate: [tex]\( -5 - 2.64 \geq -100 \Rightarrow -7.64 \geq -100 \)[/tex] (True)
- [tex]\( y \)[/tex]-coordinate: [tex]\( 0 - 2.64 \geq -100 \Rightarrow -2.64 \geq -100 \)[/tex] (True)
Since both conditions are true, Circle C lies completely within the third quadrant.
### Circle D: [tex]\((x+3)^2 + (y+9)^2 = 82\)[/tex]
- Center: The equation [tex]\((x+3)^2 + (y+9)^2 = 82\)[/tex] represents a circle with center [tex]\((-3, -9)\)[/tex].
- Radius: The radius is [tex]\(\sqrt{82} \approx 9.06\)[/tex].
For the circle to lie completely within the third quadrant:
- [tex]\( x \)[/tex]-coordinate: [tex]\( -3 - 9.06 \geq -100 \Rightarrow -12.06 \geq -100 \)[/tex] (True)
- [tex]\( y \)[/tex]-coordinate: [tex]\( -9 - 9.06 \geq -100 \Rightarrow -18.06 \geq -100 \)[/tex] (True)
Since both conditions are true, Circle D lies completely within the third quadrant.
### Conclusion:
All the circles A, B, C, and D lie completely within the third quadrant. Therefore, the correct answer is:
[tex]\[ \boxed{1, 2, 3, 4} \][/tex]
