Final answer:
To find the standard form of the equation of a circle given endpoints of a diameter, we can find the midpoint of the diameter to determine the center of the circle. The radius can be found using the distance formula between the center and one of the endpoints. Plugging the center and radius into the standard form equation (x - h)^2 + (y - k)^2 = r^2 gives the equation of the circle.
Step-by-step explanation:
To find the standard form of the equation of a circle, we need to find the center and radius of the circle. We can start by finding the midpoint of the diameter, which is the average of the x-coordinates and the average of the y-coordinates. Using the endpoints (-10,5) and (3,1), we find that the midpoint is ((-10 + 3)/2, (5 + 1)/2) = (-3.5, 3). This is the center of the circle, (h, k).
Next, we need to find the radius. We can use the distance formula to find the distance between the center and one of the endpoints of the diameter. Using the center (-3.5, 3) and the endpoint (-10, 5), we get the radius r = √((-3.5 - -10)^2 + (3 - 5)^2) = √((-3.5 + 10)^2 + (3 - 5)^2) = √((6.5)^2 + (-2)^2) = √(42.25 + 4) = √46.25.
Now we have the center (h, k) = (-3.5, 3) and the radius r = √46.25. Plugging these values into the standard form equation of a circle (x - h)^2 + (y - k)^2 = r^2, we get the equation (-3.5 - h)^2 + (y - 3)^2 = 46.25.