To determine which circles have their centers in the third quadrant, we need to identify the coordinates of the centers of each circle and then check if those coordinates satisfy the condition for being in the third quadrant. The third quadrant is defined by both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates being negative.
Let's analyze each equation one by one:
Circle A: [tex]\((x+9)^2 + (y+12)^2 = 36\)[/tex]
- This is the equation of a circle in the form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
- For Circle A, the center is [tex]\((-9, -12)\)[/tex].
- Both coordinates [tex]\(-9\)[/tex] and [tex]\(-12\)[/tex] are negative, so the center is in the third quadrant.
Circle B: [tex]\((x+14)^2 + (y-14)^2 = 84\)[/tex]
- The center is [tex]\((-14, 14)\)[/tex].
- The x-coordinate [tex]\(-14\)[/tex] is negative, but the y-coordinate [tex]\(14\)[/tex] is positive.
- Thus, the center is not in the third quadrant.
Circle C: [tex]\((x+16)^2 + (y+3)^2 = 17\)[/tex]
- The center is [tex]\((-16, -3)\)[/tex].
- Both coordinates [tex]\(-16\)[/tex] and [tex]\(-3\)[/tex] are negative, so the center is in the third quadrant.
Circle D: [tex]\((x+3)^2 + (y-6)^2 = 44\)[/tex]
- The center is [tex]\((-3, 6)\)[/tex].
- The x-coordinate [tex]\(-3\)[/tex] is negative, but the y-coordinate [tex]\(6\)[/tex] is positive.
- Thus, the center is not in the third quadrant.
Hence, the circles that have their centers in the third quadrant are:
- Circle A
- Circle C
Thus, the correct options are:
A. and C.
