Question

Find the center and radius of the sphere,

x^2+y^2+z^2+2x + 8y - 6z + 17 - 0
A. Center: (-1, -4, 3); Radius: √30
B. Center: (1, 7, -3); Radius: 4
C. Center: (-2, -8, 6); Radius: 17
D. Center: (0, 0, 0); Radius: 2

Answer 1

The center of the given sphere is (-1, -4, 3) and the radius is √17.

Step-by-step explanation:

The given equation of the sphere is:
x^2 + y^2 + z^2 + 2x + 8y - 6z + 17 = 0

To find the center of the sphere, we need to complete the square for the x, y, and z variables.

Completing the square for x gives: x^2 + 2x = (x + 1)^2 - 1

Completing the square for y gives: y^2 + 8y = (y + 4)^2 - 16

Completing the square for z gives: z^2 - 6z = (z - 3)^2 - 9

Now, substituting these values back into the equation:
(x + 1)^2 - 1 + (y + 4)^2 - 16 + (z - 3)^2 - 9 + 17 = 0

Simplifying the equation:
(x + 1)^2 + (y + 4)^2 + (z - 3)^2 = 17

Comparing this to the standard form equation of a sphere:
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2

We can determine that the center of the sphere is (-1, -4, 3) and the radius is √17.