We have ax + by + cz = 0 bx + cy + az = 0 cx + ay + bz = 0 Δ = |
a
b
c
b
c
a
c
a
b
| = - (a + b + c) (a2+b2+c2 - ab - bc - ac) = −
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(a + b + c) [(a−b)2+(b−c)2+(c−a)2] (A)→(R) a + b + c ≠0 a2+b2+c2 = ab + bc + ca ⇒ a = b = c ≠0 which implies that the equations represent identical planes. (B)→(Q) a + b + c = 0 and a2+b2+c2 ≠ab + bc + ca ⇒ Δ = 0 which implies that the equations has infinitely many solution and x = y = z = λ ∊ R satisfy the given equation. (C)→(P) a + b + c ≠0 a2+b2+c2 ≠ab + bc + ca ⇒ Δ ≠0 That is the equations has a unique solution. (D)→(S) a + b + c = 0 a2+b2+c2 = ab + bc + ac ⇒ a = b = c = 0 That is, all values of (x, y, z) satisfy the given equations and the equations represent whole of the three-dimensional space.