Which of the following circles lie completely within the third quadrant?
Check all that apply.
A. $(x+7)^2+(y+7)^2=4$
B. $(x+3)^2+(y+9)^2=82$
C. $(x+12)^2+(y+9)^2=9$
D. $(x+5)^2+(y+0)^2=7
Answer (2)
Analyze each circle's center and radius.
Check if the center's coordinates are negative (third quadrant condition).
Verify if the absolute values of the center's coordinates are greater than the radius.
Conclude that circles A and C lie completely within the third quadrant: A , C .
Explanation
Problem Analysis Let's analyze each circle to determine if it lies completely within the third quadrant. The third quadrant is defined by x < 0 and y < 0 . A circle lies completely within the third quadrant if its center has negative x and y coordinates, and the distance from the center to the x and y axes is greater than the radius.
Circle A Analysis Circle A: ( x + 7 ) 2 + ( y + 7 ) 2 = 4
Center: ( − 7 , − 7 )
Radius: 2 Since the center's x and y coordinates are negative, we check if the absolute values of the coordinates are greater than the radius:
2"> ∣ − 7∣ = 7 > 2 (True)
2"> ∣ − 7∣ = 7 > 2 (True) Therefore, circle A lies completely within the third quadrant.
Circle B Analysis Circle B: ( x + 3 ) 2 + ( y + 9 ) 2 = 82
Center: ( − 3 , − 9 )
Radius: 82 ≈ 9.06 Since the center's x and y coordinates are negative, we check if the absolute values of the coordinates are greater than the radius:
∣ − 3∣ = 3 < 82 ≈ 9.06 (False)
∣ − 9∣ = 9 < 82 ≈ 9.06 (False) Therefore, circle B does not lie completely within the third quadrant.
Circle C Analysis Circle C: ( x + 12 ) 2 + ( y + 9 ) 2 = 9
Center: ( − 12 , − 9 )
Radius: 3 Since the center's x and y coordinates are negative, we check if the absolute values of the coordinates are greater than the radius:
3"> ∣ − 12∣ = 12 > 3 (True)
3"> ∣ − 9∣ = 9 > 3 (True) Therefore, circle C lies completely within the third quadrant.
Circle D Analysis Circle D: ( x + 5 ) 2 + ( y + 0 ) 2 = 7
Center: ( − 5 , 0 )
Radius: 7 ≈ 2.65 The center lies on the x-axis ( y = 0 ), so it cannot lie completely within the third quadrant.
Conclusion Based on the analysis, circles A and C lie completely within the third quadrant.
Examples
Understanding quadrants and circles is useful in various real-world applications. For example, in radar systems, the location of an object can be determined using polar coordinates, which rely on the concept of quadrants. Similarly, in computer graphics, determining which objects are within a certain region of the screen involves quadrant analysis. This knowledge is also applicable in fields like astronomy, where the positions of celestial bodies are often described using coordinate systems.
Circles A and C lie completely within the third quadrant, while circles B and D do not. Circle A has a center at (-7, -7) and a radius of 2, and Circle C has a center at (-12, -9) and a radius of 3. Thus, the answer is A , C .
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