Question

If $a+b+c=0$ and $a^{2}+b^{2}+c^{2}-ab-bc-ca\neq 0, \, \forall a,b,c\in R$ , 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

Answer 1

Answer: (B)

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