If the vectors $a ¯ i + ¯ j + ¯ ¯ ¯ k, ¯ i + b ¯ j + ¯ ¯ ¯ k, ¯ i + ¯ j + c ¯ ¯ ¯ k$ ( $a, b, c$ are not equal to $1$ ) are coplanar, then
$\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = . . . . .$

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Solution

The correct option is D $1$
$\begin{matrix} ∣ ∣ ∣ ∣ \begin{matrix} a & 1 & 1 1 & b & 1 1 & 1 & c \end{matrix} ∣ ∣ ∣ ∣ = 0 c_{1} \to c_{1} - c_{2} c_{2} \to c_{2} - c_{3} ∣ ∣ ∣ ∣ \begin{matrix} a - 1 & 0 & 1 1 - b & b - 1 & 1 0 & 1 - c & c \end{matrix} ∣ ∣ ∣ ∣ = 0 c (a - 1) (b - 1) + (1 - b) (1 - c) - (1 - c) (a - 1) = 0 \frac{c}{1 - c} + \frac{1}{1 - a} + \frac{1}{1 - b} = 0 \frac{c}{1 - c} + 1 + \frac{1}{1 - a} + \frac{1}{1 - b} = 1 \frac{1}{1 - c} + \frac{1}{1 - a} + \frac{1}{1 - b} = 1 o p t i o n D i s c o r r e c t a n s w e r . \end{matrix}$

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