Final answer:
The equation of the circle with endpoints of the diameter at (4,4) and (-2,10) is (x - 1)² + (y - 7)² = 72, with center at (1, 7) and radius 6√2.
Step-by-step explanation:
To write the equation of a circle when you know the endpoints of the diameter, you'll first need to find the center of the circle by calculating the midpoint of the diameter's endpoints.
After that, compute the radius by finding the distance between the center and one of the endpoints.
Finally, use the circle's equation (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). With the given points (4, 4) and (-2, 10), the midpoint (center of the circle) is ((4 - 2)/2, (4 + 10)/2) which simplifies to (1, 7).
Next, we calculate the radius using the distance formula √[(x2 - x1)² + (y2 - y1)²]. This gives us √[(-2 - 4)² + (10 - 4)²] = √[36 + 36] = √72 = 6√2.
Therefore, the equation of the circle with a center at (1, 7) and radius 6√2 is (x - 1)² + (y - 7)² = (6√2)², which simplifies to (x - 1)² + (y - 7)² = 72.