Question

If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ $(a\ne b\ne c\ne 1)$ are coplanar , then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

• A. $1$
• B. $0$
• C. $2$
• D. none of these

Answer 1

Answer: (A)

$1$

$\begin{array}{c}\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=0\\ C_1\to C_1-C_2{C}_2\to C_2-C_3\\ \left|\begin{array}{ccc}a-1& 0& 1\\ 1-b& b-1& 1\\ 0& 1-c& c\end{array}\right|=0\\ c\left(a-1\right)\left(b-1\right)+\left(1-b\right)\left(1-c\right)-\left(1-c\right)\left(a-1\right)=0\\ \frac{c}{1-c}+\frac{1}{1-a}+\frac{1}{1-b}=0\\ \frac{1}{1-c}+\frac{1}{1-a}+\frac{1}{1-b}=1\\ Hence,\\ optionAiscorrectanswer.\end{array}$