The answer is D. None of these.
The circle does not intersect the y-axis because its center is 4 units away from the y-axis, which is greater than the radius of the circle.
How can you solve for the circle if it intersects?
The diameter of the circle passes through the midpoint of the line segment connecting the two given points. Therefore, the center of the circle lies at the midpoint of the line segment joining (-4, 3) and (12, -1).
Midpoint coordinates: (x_center, y_center) = ((-4 + 12) / 2, (3 - 1) / 2) = (4, 1)
The radius of the circle is half the length of the diameter, which is equal to the distance between the two given points.
Radius = √((12 - (-4))² + (-1 - 3)²) / 2 = √(82 + 22) / 2 = √68 / 2
The circle intercepts the y-axis only if the distance between the center and the y-axis is less than or equal to the radius.
Distance to y-axis = |x_center| = 4
Since 4 > √68 / 2, the circle does not intercept the y-axis.
Therefore, the answer is D. None of these.
The circle does not intersect the y-axis because its center is 4 units away from the y-axis, which is greater than the radius of the circle.