Final answer:
The equation of a circle with the given diameter endpoints is calculated by finding the midpoint, which will be the center, and the distance between the endpoints, which will give the radius. The final equation of the circle is: (x + 5)^2 + (y - 4)^2 = 5.
Step-by-step explanation:
The equation of a circle with diameter endpoints at (-4,2) and (-6,6) can be obtained by finding the midpoint (which will be the center) and the distance between the endpoints (which will give the diameter, and half of it will be the radius).
First, calculate the midpoint (h, k) of the diameter using the midpoint formula:
(h, k) = ((x_1 + x_2)/2, (y_1 + y_2)/2)
We have the endpoints (x_1, y_1) = (-4, 2) and (x_2, y_2) = (-6, 6), so:
(h, k) = ((-4 - 6)/2, (2 + 6)/2) = (-5, 4)
Next, calculate the radius r by finding the distance between the two endpoints:
r = √((x_2 - x_1)^2 + (y_2 - y_1)^2)/2
r = √((-6 + 4)^2 + (6 - 2)^2)/2 = √(4 + 16)/2 = √20/2 = √5
Now, use the standard equation of a circle with center (h, k) and radius r: (x - h)^2 + (y - k)^2 = r^2. Substituting (h, k) and r gives us:
(x + 5)^2 + (y - 4)^2 = 5
So, the final equation of the circle is: (x + 5)^2 + (y - 4)^2 = 5.