Question

If the vectors $\overrightarrow{\alpha }=a\hat{i}+\hat{j}+\hat{k},\overrightarrow{\beta }=\hat{i}+b\hat{j}+\hat{k}\text{and}\overrightarrow{\gamma }=\hat{i}+\hat{j}+c\hat{k}$ are coplanar where $a\ne 1,b\ne 1,c\ne 1$, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ equals

• A. 1.00
• B. 1
• C. 1.0

Answer 1

Answer: (A), (B), (C)

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

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

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

$\Rightarrow (a-1)(b-1)c+(c-1)(a-1)+(1-b)(1-c)=0$

$\Rightarrow \frac{c}{c-1}+\frac{1}{b-1}+\frac{1}{a-1}=0$

$\Rightarrow \frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=-1$

$\Rightarrow \frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$