Final answer:
By finding the center of the circle at (-1,1) and using the circle equation with the given y-value of 4, we find that the x-values for points on the circle are 3 and -5.
Step-by-step explanation:
The diameter of a circle has endpoints at (4,1) and (-6,1), which means the diameter is horizontal and its length is 4 - (-6) = 10 units. To find the center of the circle, we average the x-values, (4 + (-6))/2 = -1, and since the y-values are the same, the center is at (-1, 1). The radius is half the diameter, hence, r = 10/2 = 5. We use the circle equation (x + 1)^2 + (y - 1)^2 = 5^2 to find the x-values for a y-value of 4.
Substituting y = 4 into the equation: (x + 1)^2 + (4 - 1)^2 = 25 which simplifies to (x + 1)^2 + 9 = 25, hence (x + 1)^2 = 16. Taking square roots gives x + 1 = ±4, so x = 3 or x = -5. The correct answer choices containing the x-values are A) -5 and B) 3, corresponding to points on the circle where y = 4.