Write the equation of the sphere in standard form.
2x² + 2y² + 2z² = 12x - 24z + 1

Solution:

The standard equation of a sphere is (x - a)² + (x - b)² + (x - c)² = r²

Where a, b, c is the centre and r is the radius

Given: 2x² + 2y² + 2z² = 12x - 24z + 1

It can be written as

2x² + 2y² + 2z² - 12x + 24z - 1 = 0

By combining the x, y and z terms

(2x² - 12x) + 2y² + (2z² + 16z) = 1

Divide the entire equation by 2

(x² - 6x) + y² + (z² + 8z) = 1/2

Now complete the square for x by adding 9 on both sides

Complete the square for z by adding 16 on both sides

(x² - 6x + 9) + y² + (z² - 8z + 16) = 1/2 + 9 + 16

(x - 3)² + y² + (z - 4)² = 25½

Therefore, the equation of the sphere in standard form is (x - 3)² + y² + (z - 4)² = 25½.

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Write the equation of the sphere in standard form.
2x² + 2y² + 2z² = 12x − 24z + 1

Summary:

The standard form of the sphere 2x² + 2y² + 2z² = 12x - 24z + 1 is (x - 3)² + y² + (z - 4)² = 25½.