Final answer:
To find the equation of a circle, we need to find the center and radius. The center can be found by finding the midpoint of the diameter, which is the average of the x-coordinates and y-coordinates. The radius can be found using the distance formula between one endpoint and the center. The equation of the circle can then be written using the center and radius.
Step-by-step explanation:
To find the equation of a circle, we need the coordinates of the center and the radius. Since (-4, -8) and (-10, -12) are the endpoints of a diameter, we can find the center by finding the midpoint of the diameter. The midpoint is the average of the x-coordinates and the average of the y-coordinates. So, the center is ( (-4 + -10)/2, (-8 + -12)/2 ) = (-7, -10).
To find the radius, we can use the distance formula to calculate the distance between one of the endpoints and the center. Let's use (-4, -8).
The radius (r) is equal to the square root of [(x2 - x1)^2 + (y2 - y1)^2].
So, the radius is r = sqrt[(-4 - -7)^2 + (-8 - -10)^2] = sqrt[9 + 4] = sqrt[13].
The equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. Substituting the values we found, the equation of the circle is:
(x - -7)^2 + (y - -10)^2 = sqrt[13]^2
Simplifying further:
(x + 7)^2 + (y + 10)^2 = 13