Final answer:
To find the equation of a circle with given diameter endpoints, calculate the midpoint for the center, use the distance formula for the radius, and apply the circle equation formula. None of the provided options match the correct equation, which is (x - 7)^2 + (y + 3)^2 = 37.
Step-by-step explanation:
To find the equation of a circle given the endpoints of its diameter, we should first determine the circle's center point and its radius. The center point of the circle (h, k) is the midpoint of the diameter, which you can obtain by averaging the x-coordinates and the y-coordinates of the given endpoints (8, -9) and (6, +3), resulting in (7, -3). The distance between the center and one of the endpoints gives the radius of the circle, and applying the distance formula yields the radius r = √((8-7)^2 + (-9+3)^2) = √(1^2 + (-6)^2) = √(1+36) = √37.
The equation of a circle is given by the formula (x - h)^2 + (y - k)^2 = r^2, so after substituting the values for h, k, and r, the equation becomes (x - 7)^2 + (y + 3)^2 = 37. Checking the given options, none of them exactly matches the derived equation, which means there may have been a typo in the options provided. However, to answer the step-by-step approach, the correct equation is not listed.