Answer:
(x + 1)^2 + y^2 = 64
Explanation:
Step 1: Find the center of the circle (h, k) by taking the average of the x-coordinates and the y-coordinates of the endpoints of the diameter:
Center (h, k) = ((2 + (-6)) / 2, (-4 + 4) / 2)
Center (h, k) = (-2 / 2, 0 / 2)
Center (h, k) = (-1, 0)
So, the center of the circle is (-1, 0).
Step 2: Calculate the radius of the circle (r), which is half the length of the diameter. Use the distance formula between the two endpoints of the diameter:
Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
Diameter length = √((-6 - 2)^2 + (4 - (-4))^2)
Diameter length = √((-8)^2 + (8)^2)
Diameter length = √(64 + 64)
Diameter length = √128
Radius (r) = Diameter length / 2
Radius (r) = √128 / 2
Radius (r) = √64
Radius (r) = 8
Step 3: Now that we have the center (h, k) and the radius (r), we can use the standard form of the equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
Substitute the values of the center and radius into the equation:
(x - (-1))^2 + (y - 0)^2 = 8^2
(x + 1)^2 + y^2 = 64
So, the equation of the circle whose diameter has endpoints at (2, -4) and (-6, 4) is:
(x + 1)^2 + y^2 = 64