Answer:
the center of the sphere is (1, -2, 3) and the radius is 2 units.
Explanation:
The equation x^2 + y^2 + z^2 - 2x + 4y - 6z + 10 = 0 represents a sphere in three-dimensional space. To find its center and radius, we need to rewrite the equation in standard form.
First, let's rearrange the equation by completing the square for each variable.
For the x terms:
x^2 - 2x = (x^2 - 2x + 1) - 1 = (x - 1)^2 - 1.
For the y terms:
y^2 + 4y = (y^2 + 4y + 4) - 4 = (y + 2)^2 - 4.
For the z terms:
z^2 - 6z = (z^2 - 6z + 9) - 9 = (z - 3)^2 - 9.
Substituting these results into the equation, we have:
(x - 1)^2 - 1 + (y + 2)^2 - 4 + (z - 3)^2 - 9 + 10 = 0.
Simplifying, we get:
(x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 4.
Comparing this with the standard form equation of a sphere:
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2,
we can see that the center of the sphere is (1, -2, 3) and the radius is 2.