Find the center and radius of the sphere x² + y² + z² - 6y + 8z = 0.

Solution:

We have equation of the sphere centred at (h, k. l) and having radius r is given by 

(x - h)² + (y - k)² + (z - l)² = r²---------------------------(1)

To identify the centre and radius of the given sphere we have to convert the given equation

 x² + y² + z² - 6y + 8z = 0 to the form in equation (1) 

Now group the x , y and z terms and by completing the square, we get

x² + (y² - 6y + 9) +(z² + 8z + 16) = 9 + 16

(x - 0)² + (y - 3)² + (z + 4)² = 5²

Hence centre of the sphere is (0, 3, -4) and radius = 5

Find the center and radius of the sphere x² + y² + z² - 6y + 8z = 0.

Summary:

The center and the radius of the sphere with the equation, x² + y² + z² - 6y + 8z = 0 are (0, 3, -4) and 5 respectively.