The first step in finding the equation of the circle is to find the coordinates of the center, which is the midpoint of the line connecting the two endpoints of the diameter. These endpoints are (-2, -6) and (2, -2).
To find the coordinates of the center of the circle, we average the x-coordinates of the endpoints and the y-coordinates of the endpoints.
To find the x-coordinate of the center, we add -2 and 2 and divide by 2 to get 0.
To find the y-coordinate of the center, we add -6 and -2 and divide by 2 to get -4.
Therefore, the center of the circle is located at (0, -4).
Next, we need to find the radius of the circle. To do this, we calculate the distance between the center of the circle and one of the endpoints of the diameter.
We can use the distance formula, which is:
sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we'll use the coordinates of the center and one endpoint of the diameter. So, our formula will look like this:
sqrt((2 - 0)^2 + (-2 - (-4))^2), which simplifying results in sqrt(8) or 2.8284271247461903.
So, the radius of the circle is approximately 2.8284271247461903 units.
Now we have the center and the radius, we can write the equation of the circle in standard form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle, and r is the radius.
Substitute the center (0, -4) and the radius 2.8284271247461903 into the equation to get:
(x - 0)^2 + (y - (-4))^2 = (2.8284271247461903)^2
So, the equation of the circle whose diameter has endpoints (-2, -6) and (2, -2) is:
x^2 + (y + 4)^2 = 7.999999999999999.