To find the radius of the sphere that is concentric with the given sphere \( S \) and passes through the origin, we must first address key concepts related to the equation of a sphere.

The general equation of a sphere is:

\(x^2 + y^2 + z^2 + 2gx + 2fy + 2hz + c = 0\)

In this equation, \((h, k, l)\) is the center of the sphere, where \( h = -g \), \( k = -f \), and \( l = -h \). The radius \( r \) is given by:

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

Given the equation of the sphere \( S \):

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

We can equate this to the general form to identify:

• \(2g = -4 \Rightarrow g = -2\)
• \(2f = -6 \Rightarrow f = -3\)
• \(2h = -12 \Rightarrow h = -6\)

Thus, the center of sphere \( S \) is \((-g, -f, -h) = (2, 3, 6)\).

Since the new sphere is concentric with \( S \), it also has the center \((2, 3, 6)\). This sphere passes through the origin \((0, 0, 0)\). Therefore, the radius \( R \) of the new sphere is the distance from the origin to the center of the sphere:

\(R = \sqrt{(2 - 0)^2 + (3 - 0)^2 + (6 - 0)^2}\)

Calculating this gives:

\(R = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7\)

Therefore, the radius of the sphere that is concentric with \( S \) and passes through the origin is 7.