Question

Write the equation of the sphere in standard form. x²+ y²+ z²+ 6x − 2y − 2z = 5 find its center and radius. center (x, y, z) = radius

Answer 1

The equation of the sphere in standard form is (x+3)² + (y-1)² + (z-1)² = 16 with the center at (-3, 1, 1) and the radius of 4.

Step-by-step explanation:

The equation of a sphere in standard form is (x-h)² + (y-k)² + (z-l)² = r², where (h,k,l) are the coordinates of the sphere's center and r is the radius. We complete the square for the given equation x²+ y²+ z²+ 6x − 2y − 2z = 5 to bring it to standard form. Group the x, y, and z terms, and factor them to complete the squares:

(x² + 6x + 9) - 9 + (y² - 2y + 1) - 1 + (z² - 2z + 1) - 1 = 5

Rewriting, we have:

(x+3)² + (y-1)² + (z-1)² = 5 + 9 + 1 + 1

Which simplifies to:

(x+3)² + (y-1)² + (z-1)² = 16

Thus, the center of the sphere is (-3, 1, 1) and the radius is 4.