Step 1: Understanding the Concept:
The radius of a sphere is the distance from its center to any point on its surface. If the sphere passes through the origin, then the origin is a point on its surface.
Step 2: Key Formula or Approach:
The distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3D space is given by the distance formula:
\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \]
We will use this formula to find the distance between the center of the sphere and the origin. This distance will be the radius.
Step 3: Detailed Explanation:
The center of the sphere is given as \(C = (4, 4, -2)\).
The sphere passes through the origin, which is the point \(O = (0, 0, 0)\).
The radius \(r\) is the distance between C and O.
Using the distance formula:
\[ r = \sqrt{(4-0)^2 + (4-0)^2 + (-2-0)^2} \]
\[ r = \sqrt{4^2 + 4^2 + (-2)^2} \]
\[ r = \sqrt{16 + 16 + 4} \]
\[ r = \sqrt{36} \]
\[ r = 6 \]
Step 4: Final Answer:
The radius of the sphere is 6, which corresponds to option (D).