Complete the square to write the equation of the sphere in standard form. Find the center and radius. x² + y² + z² - 2x + 6y + 8z + 1 = 0

Solution:

The equation of the sphere in the standard form is given below:

(x - x₀)² + (y - y₀)² + (z - z₀)² = a²

Let us convert the given equation of the sphere into the standard form.

x² + y² + z² - 2x + 6y + 8z + 1 = 0

(x² - 2x + (1)²) + (y² + 6y + (3)² - 9) + z² + 8z + (4)² - 16 = 0

(x - 1)² + (y + 3 )² + (z +4)² =  9 + 16

(x - 1)² + (y + 3 )² + (z +4)² = 25

The center coordinates are x₀ = 1; y₀ = -3; z₀ = -4 and radius a = 5

Complete the square to write the equation of the sphere in standard form. Find the center and radius. x² + y² + z² - 2x + 6y + 8z + 1 = 0

Summary:

The equation of the sphere in standard form is (x - 1)² + (y + 3 )² + (z +4)² = 25. The center is (1, -3, -4) and radius is 5.