Question

Write the equation of the sphere in standard form 2x2 2y2 2z2 = 4x − 24z 1

Answer 1

The equation of the sphere in standard form is
`\(x² + y² + z² - 4x + 24z - 1 = 0\)`.

Step-by-step explanation:

To express the given equation in standard form we gather all the terms on one side to set it equal to zero. Start by completing the square for the x and z terms by halving their coefficients and squaring. The equation becomes
`\(x² - 4x + y²+ z² - 24z = 1\)`.

Now add constants to complete the square for both x and z resulting in
`\((x² - 4x + 4) + y² + (z² - 24z + 144) = 1 + 4 + 144\)`. Simplify this to get
`\((x - 2)² + y² + (z - 12)² = 149\)` which is the standard form of the sphere equation.

Understanding the process of completing the square in multivariable equations helps in transforming them to standard form a crucial step in analyzing and interpreting geometric shapes represented by these equations.