Final answer:
To find the equation of a circle given the endpoints of a diameter, use the midpoint formula to find the center and the distance formula to find the radius. Then, write the equation of the circle as (x - h)^2 + (y - k)^2 = r^2.
Step-by-step explanation:
To find the equation of a circle given the endpoints of a diameter, we need to find the center of the circle and the radius. The center of the circle is the midpoint of the diameter, which can be found using the midpoint formula: [(x1 + x2)/2, (y1 + y2)/2]. In this case, the midpoint is [(6 - 2)/2, (-13 + 15)/2] = [2, 1].
The radius of the circle is half the length of the diameter. Using the distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2), we can find the length of the diameter. In this case, the diameter is sqrt((6 - (-2))^2 + (-13 - 15)^2) = sqrt(64 + 784) = sqrt(848) = 4sqrt(53).
Therefore, the equation of the circle can be written as (x - 2)^2 + (y - 1)^2 = (2sqrt(53))^2 or (x - 2)^2 + (y - 1)^2 = 212.