Question

If a, b, c are non zero real numbers and if the equations $(a-1)x=y+z,(b-1)y=z+x,(c-1)z=(x+y)$ has a non trivial solution then ab + bc + ca equals

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

Answer 1

Answer: (B)

abc

Given system of equations has non-trivial solution.

$\Rightarrow \left|\begin{array}{ccc}1-a& 1& 1\\ 1& 1-b& 1\\ 1& 1& 1-c\end{array}\right|=0$

$\Rightarrow (1-a)\left[\right(1-b\left)\right(1-c)-1]+c+b=0$

$\Rightarrow 1-(a+b+c)+ab+bc+ca-abc-1+(a+b+c)=0$

$\therefore ab+bc+ca=abc$

Hence, option B.