Finding Sphere Constant k from Radius

The general equation of a sphere is given as:

\(x^2 + y^2 + z^2 - 4x - 6y - 12z + k = 0\)

The standard form of a sphere's equation is \(x^2 + y^2 + z^2 + 2gx + 2fy + 2hz + d = 0\), where the center is \((-g, -f, -h)\) and the radius \(r\) is calculated using the formula:

\(r = \sqrt{g^2 + f^2 + h^2 - d}\)

Comparing Coefficients

By comparing the given equation with the standard form, we identify the coefficients:

• \(2g = -4 \implies g = -2\)
• \(2f = -6 \implies f = -3\)
• \(2h = -12 \implies h = -6\)
• \(d = k\)

Calculating Radius and Solving for k

We are given that the radius \(r = 8\) units.

Substitute the values of \(g\), \(f\), \(h\), \(d\), and \(r\) into the radius formula:

\(8 = \sqrt{(-2)^2 + (-3)^2 + (-6)^2 - k}\)

Calculate the squares:

\(8 = \sqrt{4 + 9 + 36 - k}\)

Simplify the terms under the square root:

\(8 = \sqrt{49 - k}\)

Square both sides of the equation to eliminate the square root:

\(8^2 = 49 - k\)

\(64 = 49 - k\)

Rearrange the equation to solve for \(k\):

\(k = 49 - 64\)

\(k = -15\)

Therefore, the value of \(k\) is -15.