# Write the equation of the sphere 2x^{2} + 2y^{2} + 2z^{2} = 8x - 24z + 1 in standard form. Find its center and radius.

**Solution:**

From the question it is given that 2x^{2} + 2y^{2} + 2z^{2} = 8x - 24z + 1

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

Then, completing squares in the equation we get,

= 2(x^{2} - 4x + 4) + 2y^{2} + 2(x^{2} + 12x + 36) = 1 + 8 + 72

= 2(x - 2)^{2} + 2y^{2} + 2(z + 6)^{2} = 81

By cross multiplication we get,

= (x - 2)^{2} + y^{2} + (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 2x^{2} + 2y^{2} + 2z^{2} = 8x - 24z + 1 in standard form is (x - 2)^{2} + y^{2} + (z + 6)^{2} = 81/2. Center is (2, 0, -6) and radius is 9/√2.

## Write the equation of the sphere 2x^{2} + 2y^{2} + 2z^{2}= 8x - 24z + 1 in standard form. Find its center and radius.

**Summary:**

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