Final answer:
To find the equation of the circle with endpoints of a diameter at (7,6) and (-5,8), calculate the midpoint to determine the center and use the distance formula to find the radius. The equation in standard form is (x - 1)^2 + (y - 7)^2 = 37.
Step-by-step explanation:
The standard form of the equation of a circle is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
To find the center (h, k) of the circle with endpoints of a diameter at (7,6) and (-5,8), we calculate the midpoint of the line segment joining these endpoints. Using the midpoint formula, we get h = (7 - 5) / 2 and k = (6 + 8) / 2, which simplifies to (1, 7).
To find the radius r, we calculate the distance between the center and one of the endpoints using the distance formula. The distance or the radius r is found to be sqrt(6^2 + (-1)^2), which simplifies to sqrt(37).
Therefore, the standard form of the equation of the circle is (x - 1)^2 + (y - 7)^2 = 37.