Question

The length of the perpendicular drawn from the point $(2,1,4)$ to the plane containing the lines $\overrightarrow{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}-\hat{k})$ and $\overrightarrow{r}=(\hat{i}+\hat{j})+\mu (-\hat{i}+\hat{j}-2\hat{k})$ is :

• A. $3$
• B. $\sqrt{3}$
• C. $\frac{1}{3}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

The correct option is B $\sqrt{3}$

Equation of plane containing lines $\overrightarrow{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}-\hat{k})$ and $\overrightarrow{r}=(\hat{i}+\hat{j})+\mu (-\hat{i}+\hat{j}-2\hat{k})$ is given by

$\left|\begin{array}{ccc}x-1& y-1& z\\ 1& 2& -1\\ -1& 1& -2\end{array}\right|=0$

$\Rightarrow -x+y+z=0$
Distance from the point $(2,1,4)$ is
$\left|\frac{-2+1+4}{\sqrt{(-1{)}^2+1^2+1^2}}\right|=\sqrt{3}$