To represent a circle with a diameter of 12 units and a center that lies on the y-axis, we can use the standard form of the equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle, and r is the radius.
If the center of the circle lies on the y-axis, then the x-coordinate of the center is 0. Also, since the diameter is 12 units, the radius is 6 units.
Using these values, we can eliminate the equations that do not meet these conditions:
- x2 + (y – 3)2 = 36: This circle has a center at (0, 3), which is not on the y-axis.
- x2 + (y – 5)2 = 6: This circle has a center at (0, 5), which is not on the y-axis.
- (x – 4)² + y² = 36: This circle has a center at (4, 0), which is not on the y-axis.
- (x + 6)² + y² = 144: This circle has a center at (-6, 0), which is not on the y-axis.
- x2 + (y + 8)2 = 36: This circle has a center at (0, -8), which is on the y-axis.
Therefore, the two equations that represent circles with a diameter of 12 units and a center that lies on the y-axis are:
- x2 + (y + 8)2 = 36
- x2 + (y - 8)2 = 36
Note that the second equation is also valid, since the center of the circle can also be located at (0, -8).