a,b,c are distinct real numbers, not equal to one. If ax+y+z=0,x+by+z=0, and x+y+cz=0 have a non trivial solution, then the value of 11−a+11−b+11−c is equal to

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$a, b, c are distinct real numbers, not equal to one. If ax + y + z = 0, x + by + z = 0, and$ $x + y + cz = 0 have a non trivial solution, then the value of \frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} is equal$ to

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zero

-1

None of these

answer is B.

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Detailed Solution

Since the system has non-trivial solution, we have
$|\begin{array}{lll} a 1 1 \\ 1 b 1 \\ 1 1 c \end{array}| = 0$
$Applying R_{1} \to R_{1} - R_{2}, R_{2} \to R_{2} - R_{3}, we get$
$Δ = |\begin{array}{ccc} a - 1 & 1 - b & 0 \\ 0 & b - 1 & 1 - c \\ 1 & 1 & c \end{array}| = 0$
Or $c (1 - a) (1 - b) + (1 - b) (1 - c) - (1 - c) (a - 1) = 0$
$Dividing throughout by (1 - a) (1 - b) (1 - c), we get$
$\begin{array}{l} \frac{c}{1 - c} + \frac{1}{1 - c} + \frac{1}{1 - b} = 0 \\ Or - 1 + \frac{1}{1 - c} + \frac{1}{1 - b} + \frac{1}{1 - a} = 0 \\ \frac{1}{1 - c} + \frac{1}{1 - a} + \frac{1}{1 - b} = 1 \end{array}$