Question

Equation of the plane that contains the lines $r=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}-\hat{k})$ and $r=(\hat{i}+\hat{j})=\mu (-\hat{i}+\hat{j}-2\hat{k})$, is

• A. $r.(2\hat{i}+\hat{j}-3\hat{k})=-4$
• B. $r\times (-\hat{i}+\hat{j}+\hat{k})=0$
• C. $r.(-\hat{i}+\hat{j}+\hat{k})=0$
• D. None of these

Answer 1

Answer: (C)

$r.(-\hat{i}+\hat{j}+\hat{k})=0$

$\overrightarrow{r}=(i+j)+\lambda (i+2j-k)\overrightarrow{r}=i+j+\mu (-i+j-2k)$

Normal to the plane containing both line is $\overrightarrow{b_1}\times \overrightarrow{b_2}$

$\Rightarrow (i+2j-k)\times (-i+j-2k)\Rightarrow -3i+3j+3k$

So the equation of plane is

$\overrightarrow{r}.(-3i+3j+3k)=d$

It passes through $(1,1,0)$

$\Rightarrow (i+j).(-3i+3j+3k)=d\Rightarrow d=0$

So the equation of plane becomes

$\overrightarrow{r}.(-3i+3j+3k)=0\overrightarrow{r}.(-i+j+k)=0$