Concept:1). The general form of the equation of the sphere is x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 where u, v, w and d are constant.2). The equation of the plane whose intercepts are a, b, c on the x, y, z axes respectively is \(\displaystyle \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\), where a, b, c ? 0Calculation:The given equation of plane is 6x + ky + 3z - 12 = 0In intercept form \(\displaystyle \frac{x}{2}+\frac{y}{\frac{12}{k}}+\frac{z}{4}=1\)So the coordinates of A, B, C are (2,0,0), (0,\(\frac{12}{k}\),0) and (0,0,4) respectively.Let the equation of the sphere be x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0As the sphere passes through the origin, it satisfies point (0, 0, 0) ? d = 0.? x2 + y2 + z2 + 2ux + 2vy + 2wz = 0 ------(1)Equation (1) passes through the point A, B, C.At A (2,0,0) we get, 4 + 2u.2 = 0 ? u = -1At B (0,\(\frac{12}{k}\),0) we get, \(\frac{144}{k^2}\) + 2v.\(\frac{12}{k}\) = 0 ? v = -\(\frac{6}{k}\)At C (0,0,4) we get, 16 + 2w.4 = 0 ? w = -2So the required equation of the sphere isx2 + y2 + z2 + 2.(-1).x + 2.(-\(\frac{6}{k}\)).y + 2.(-2).z = 0? x2 + y2 + z2 - 2x + (-\(\frac{12}{k}\)).y - 4z = 0 ------(2)Comparing x2 + y2 + z2 - 2x - 3y - 4z = 0 with equation (1) we get, k = 4.? The value of k is 4.