Which of the following circles have their centers in the third quadrant?
Check all that apply.
A. $(x+14)^2+(y-14)^2=84$
B. $(x+9)^2+(y+12)^2=36$
C. $(x+3)^2+(y-6)^2=44$
D. $(x+16)^2+(y+3)^2=17
Answer (2)
Identify the center ( h , k ) of each circle from its equation.
Determine if both h and k are negative.
Circle B has center ( − 9 , − 12 ) , which is in the third quadrant.
Circle D has center ( − 16 , − 3 ) , which is in the third quadrant.
The circles with centers in the third quadrant are B and D. B , D
Explanation
Understanding the Problem The equation of a circle is given by ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle. The third quadrant is where both x and y coordinates are negative. We need to find the circles with centers in the third quadrant.
Analyzing Each Option Let's analyze each option: A. ( x + 14 ) 2 + ( y − 14 ) 2 = 84 . The center is ( − 14 , 14 ) . Since the y -coordinate is positive, this circle's center is not in the third quadrant. B. ( x + 9 ) 2 + ( y + 12 ) 2 = 36 . The center is ( − 9 , − 12 ) . Both coordinates are negative, so this circle's center is in the third quadrant. C. ( x + 3 ) 2 + ( y − 6 ) 2 = 44 . The center is ( − 3 , 6 ) . Since the y -coordinate is positive, this circle's center is not in the third quadrant. D. ( x + 16 ) 2 + ( y + 3 ) 2 = 17 . The center is ( − 16 , − 3 ) . Both coordinates are negative, so this circle's center is in the third quadrant.
Conclusion Therefore, the circles with centers in the third quadrant are B and D.
Examples
Understanding quadrants is useful in navigation and mapping. For example, if you are using a coordinate system to locate objects on a map, knowing which quadrant an object is in helps you quickly determine its general direction and position relative to the origin.
The circles with centers in the third quadrant are B and D, with centers at (-9, -12) and (-16, -3), respectively. Both coordinates of these centers are negative, placing them in the correct quadrant. A and C are not in the third quadrant due to their positive y-coordinates.
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