Write the equation of the sphere 2x² + 2y² + 2z² = 8x - 24z + 1 in standard form. Find its center and radius.

Solution:

From the question it is given that 2x² + 2y² + 2z² = 8x - 24z + 1

The equation of the sphere is of the form(x-a)² + (y-b)² +(z-c)² = r² , centered at (a,b,c) and with radius r.

Then, completing squares in the equation we get,

= 2(x² - 4x + 4) + 2y² + 2(x² + 12x + 36) = 1 + 8 + 72

= 2(x - 2)² + 2y² + 2(z + 6)² = 81

By cross multiplication we get,

= (x - 2)² + y² + (z + 6)² = 81/2

So, an equation of a sphere with a center is (2, 0, -6)

And radius is √(81/2) = 9/√2

Therefore, the equation of the sphere 2x² + 2y² + 2z² = 8x - 24z + 1 in standard form is (x - 2)² + y² + (z + 6)² = 81/2. Center is (2, 0, -6) and radius is 9/√2.

Write the equation of the sphere 2x² + 2y² + 2z²= 8x - 24z + 1 in standard form. Find its center and radius.

Summary:

The equation of the sphere 2x² + 2y² + 2z² = 8x - 24z + 1 in standard form is (x - 2)² + y² + (z + 6)² = 81/2. Center is (2, 0, -6) and radius is 9/√2.