Final answer:
The equation in standard form of the circle that has ends of a diameter at (-4,-1) and (0,3) is (x + 2)^2 + (y - 1)^2 = 8.
Step-by-step explanation:
To find the equation of a circle in standard form, we can start by finding the midpoint of the given diameter. The midpoint is the average of the x-coordinates and the average of the y-coordinates of the two endpoints.
In this case, the x-coordinate of the midpoint is (-4 + 0) / 2 = -2 and the y-coordinate of the midpoint is (-1 + 3) / 2 = 1. Using the midpoint and one of the endpoints, we can find the radius of the circle by calculating the distance between them.
The radius is the square root of the difference of the x-coordinates squared plus the difference of the y-coordinates squared. In this case, the radius is sqrt((-4 - (-2))^2 + (-1 - 1)^2) = sqrt(4 + 4) = sqrt(8). Now that we have the midpoint (-2, 1) and the radius sqrt(8), we can substitute these values into the standard form equation of a circle, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the midpoint and r is the radius. Therefore, the equation in standard form of the circle is (x + 2)^2 + (y - 1)^2 = 8.