Question

If vectors $\alpha \overline{i}+\overline{j}+\overline{k}$, $\overline{i}+\beta \overline{j}+\overline{k}$, $\overline{i}+\overline{j}+\gamma \overline{k}$ $(\alpha \ne 1,\beta \ne 1,\gamma \ne 1)$ are coplanar, then the value of $\frac{1}{1-\alpha }+\frac{1}{1-\beta }+\frac{1}{1-\gamma }$ is

• A. $0$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

Answer: (C)

$1$

Since given vectors are coplanar

$\therefore$ $\left|\begin{array}{ccc}\alpha & 1& 1\\ 1& \beta & 1\\ 1& 1& \gamma \end{array}\right|=0$

$\Rightarrow$ $\alpha \beta \gamma -(\alpha +\beta +\gamma )+2=0$
$\Rightarrow$ $\frac{1}{1-\alpha }+\frac{1}{1-\beta }+\frac{1}{1-\gamma }$

$=\frac{3-2(\alpha +\beta +\gamma )+(\alpha \beta +\beta \gamma +\gamma \alpha )}{1-(\alpha +\beta +\gamma )+(\alpha \beta +\beta \gamma +\gamma \alpha )-\alpha \beta \gamma }$
$=\frac{-1-2\alpha \beta \gamma +(\alpha \beta +\beta \gamma +\gamma \alpha )}{-1-2\alpha \beta \gamma +(\alpha \beta +\beta \gamma +\gamma \alpha )}$

$=1$