If a+b+c=0 and a2+b2+c2-ab-bc-ca≠ 0, ∀ a,b,c∈ R , then the system of equations ax+by+cz=0, bx+cy+az=0 and cx+ay+bz=0 has

Question

If a+b+c=0 and a2+b2+c2-ab-bc-ca≠ 0, ∀ a,b,c∈ R , then the system of equations ax+by+cz=0, bx+cy+az=0 and cx+ay+bz=0 has (A) A unique solution (B) Infinite solutions (C) No solution (D) Exactly two solutions

Tardigrade Admin · Accepted Answer

Note that | a b c b c a c a b |=(- (a + b + c)/2)[(a - b)2 + (b - c2) + (c - a)2] Since, D=0 (0,0 , 0) is a solution, hence the system has infinite solutions.

Q. If $a + b + c = 0$ and $a^{2} + b^{2} + c^{2} - ab - b c - c a \neq = 0, \forall a, b, c \in R$ , then the system of equations $a x + b y + cz = 0,$ $b x + cy + a z = 0$ and $c x + a y + b z = 0$ has

A unique solution

Infinite solutions

No solution

Exactly two solutions

Note that $∣ ∣ a b c b c a c a b ∣ ∣ = \frac{- ( a + b + c )}{2} [(a - b)^{2} + (b - c^{2}) + (c - a)^{2}]$
Since, $D = 0$ & $(0, 0, 0)$ is a solution, hence the system has infinite solutions.

Q. If a+b+c=0 and a2+b2+c2−ab−bc−ca=0,∀a,b,c∈R , then the system of equations ax+by+cz=0, bx+cy+az=0 and cx+ay+bz=0 has