Final answer:
The standard equation of the circle with endpoints (-7,-1) and (3,3) as the diameter is (x - (-2))^2 + (y - 1)^2 = (√29)^2.
Step-by-step explanation:
The standard equation of a circle is given by the equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
First, we need to find the center of the circle. The center of a circle is the midpoint of its diameter, which can be found by averaging the x-coordinates and the y-coordinates of the endpoints of the diameter.
The x-coordinate of the center is (-7 + 3) / 2 = -2 and the y-coordinate of the center is (-1 + 3) / 2 = 1.
So the center of the circle is (-2, 1).
Next, we need to find the radius of the circle. The radius is equal to half the length of the diameter.
The distance between the endpoints (-7,-1) and (3,3) can be found using the distance formula: √[(x2 - x1)^2 + (y2 - y1)^2].
The distance is √[(3 - (-7))^2 + (3 - (-1))^2] = √[(3 + 7)^2 + (3 + 1)^2] = √[10^2 + 4^2] = √(100 + 16) = √116 = 2√29.
The radius is half of this distance, so the radius is √29.
Putting it all together, the standard equation of the circle is (x - (-2))^2 + (y - 1)^2 = (√29)^2.