Question

Assertion :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 $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$ provided $a\ne 1,b\ne 1,c\ne 1$. Reason: Vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar, then $\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})=0$.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

Three vectors $\overline{a},\overline{b},\overline{c}$ are coplanar then
$\left[\overline{a}\overline{b}\overline{c}\right]=\overline{a}\cdot (\overline{b}\times \overline{c})$
$\therefore$ Reason is true & Correct explanation for Assertion (A)
Now , $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k},\hat{i}+\hat{j}+c\hat{k}$ are given coplanar
$\Rightarrow \left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|\left[\hat{i}\hat{j}\hat{k}\right]=0$
$\Rightarrow \left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=0as\hat{i}\cdot (\hat{j}\times \hat{k})=1$
Using $C_2\to C_2-C_1,C_3\to C_3-C_2$
$\Rightarrow \left|\begin{array}{ccc}a& 1-a& 0\\ 1& b-1& 1-b\\ 1& 0& c-1\end{array}\right|=0$
$\Rightarrow a(b-1)(c-1)-(1-a)[(c-1)-(1-b)]=0$
$\Rightarrow a(b-1)(c-1)-(1-a)(c-1)+(1-a)(1-b)=0$
on dividing by $\Rightarrow (1-a)(1-b)(1-c)=0$
$\Rightarrow \frac{1-(1-a)}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=0$
$\Rightarrow \frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$
$\Rightarrow$ Assertion (A) is true .