If `a , b , c` are non-zero real numbers and if the system of equations `(a-1)x=y=z` `(b-1)y=z+x` `(c-1)z=x+y` has a non-trivial solution, then prove that `a b+b c+c a=a b c`

Options

A. a+b+c
B. abc
C. 1
D. None of these

Answer

B

Explanation

For nontrivial solution
`|(a-1,-1,-1),(1,1-b,1),(1,1,1-c)|=0`
Applying `C_1 to C_1 -C_3` and `C_2 to C_2 -C_3` , then
`|(a,0,-1),(0,-b,1),(c,c,1-c)|=0 rArr ` a(-b+bc-c)-0+c(0-b)=0 `rArr` ab+bc+ca=abc