Question

If $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k},\hat{i}+\hat{j}+c\hat{k}$ are coplanar then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

• A. $2$
• B. $1$
• C. $-1$
• D. $3$

Answer 1

The correct option is B $1$

$a\hat{i}+\hat{j}+\hat{k}=\lambda (\hat{i}+b\hat{j}+\hat{k})+\mu (\hat{i}+\hat{j}+c\hat{k})$ (Property of coplanarity )

$\lambda +\mu =a$ --------(1)

$\lambda b+\mu =1$ --------(2)

$\lambda +c\mu =1$ --------(3)

$\lambda (b-1)=1-a$

$(c-1)\mu =1-a$

$\lambda +\mu =(a-1)(\frac{1}{1-b}+\frac{1}{1-c})$

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

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

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