Final answer:
The sphere centered at (-10, -7, 1) with radius 3 has the equation (x + 10)^2 + (y + 7)^2 + (z - 1)^2 = 9.
The circle representing the intersection with the plane z = 2 has the equation (x + 10)^2 + (y + 7)^2 = 8, and its center is (-10, -7, 2) with radius √8.
Step-by-step explanation:
The equation of a sphere centered at a point (h, k, l) with radius r is (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2.
In this case, the center is (-10, -7, 1) and the radius is 3, so the equation of the sphere is as follows:
(x + 10)^2 + (y + 7)^2 + (z - 1)^2 = 9.
To find the equation of the intersection of this sphere with the plane z = 2, we substitute z with 2 in the equation of the sphere, giving:
(x + 10)^2 + (y + 7)^2 + (2 - 1)^2 = 9.
Simplify this equation to get:
(x + 10)^2 + (y + 7)^2 = 8.
This represents a circle on the plane z = 2 with center (-10, -7, 2) and radius √8.