Equation of a sphere

Solution

Option (c) is correct.

Explanation:

The given equation of the sphere S is To find the center and radius of the sphere, we complete the square for each variable. For \(x\): \[ x^2 - 4x \rightarrow (x-2)^2 - 4 \] For \(y\): \[ y^2 - 6y \rightarrow (y-3)^2 - 9 \] For \(z\): \[ z^2 - 12z \rightarrow (z-6)^2 - 36 \] Substituting back into the equation gives: \[ (x-2)^2 + (y-3)^2 + (z-6)^2 - 4 - 9 - 36 + k = 0 \] Thus, the center of the sphere \(C\) is \((2, 3, 6)\) and the equation can be rewritten as: \[ (x-2)^2 + (y-3)^2 + (z-6)^2 = 49 - k \] This reveals that the radius of the sphere is given by: \[ r = \sqrt{49 - k} \] For the sphere to pass through the origin \((0, 0, 0)\), we use the distance formula to find the distance from the center to the origin: \[ d = \sqrt{(2 - 0)^2 + (3 - 0)^2 + (6 - 0)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \] Thus, we set the radius equal to this distance: \[ \sqrt{49 - k} = 7 \] Squaring both sides yields: \[ 49 - k = 49 \implies k = 0 \] The radius of the sphere when \(k = 0\) is: \[ \sqrt{49 - 0} = 7 \] Thus, the radius of the sphere passing through the origin and concentric with the sphere S is 7.