Find the center and radius of the sphere whose equation is given by x² + y² + z² + 4x - 2z - 8 = 0. R = ?

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 center and radius of the given sphere we have to convert the given equation x² + y² + z² + 4x - 2z - 8 = 0 to the form as in the equation (1) 

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

x² + y² + z² + 4x - 2z = 8

(x² + 4x + 4 )+ (y²) + (z² - 2z + 1)  = 8 + 4 + 1

(x + 2)² + (y - 0)² + (z - 1)² = 13

∴ (x + 2)² + (y - 0)² + (z - 1)² = (√13)²

Hence centre of the sphere is (-2, 0, 1) and radius = √13

Find the center and radius of the sphere whose equation is given by x² + y² + z² + 4x - 2z - 8 = 0. R = ?

Summary:

The center and the radius of the sphere with the equation, x² + y² + z² + 4x - 2z - 8 = 0 are (-2, 0, 1) and √13 respectively.