Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub
For real a,b,c, match the conditions for the homogeneous system:
List I | List II |
(A) a+b+c ≠ 0 and a²+b²+c² = ab+bc+ca | (P) The planes meet only at a single point. |
(B) a+b+c = 0 and a²+b²+c² ≠ ab+bc+ca | (Q) The solution set is the line x=y=z. |
(C) a+b+c ≠ 0 and a²+b²+c² ≠ ab+bc+ca | (R) The equations represent identical planes. |
(D) a+b+c = 0 and a²+b²+c² = ab+bc+ca | (S) The solution set is the whole three-dimensional space. |
Options:
Answer:
Solution:
Let M be the coefficient matrix. Use S and T below. In (A), all coefficients are equal and nonzero, giving identical planes, R. In (B), S=0 and T is nonzero. The principal minors sum to -T, so rank is 2; the nullspace is the line x=y=z, Q. In (C), the determinant is nonzero, so only the origin satisfies the system, P. In (D), all coefficients vanish, so the solution is all space, S. Thus the correct option is A.
