Question

Let $L_1$ be the line ${\overrightarrow{r}}_1=2\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k}+\lambda (\overrightarrow{i}+2\overrightarrow{k})$ and let $L_2$ be the line ${\overrightarrow{r}}_2=3\overrightarrow{i}+\overrightarrow{j}+\mu (\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k})$ . Let $\pi$ be the plane which contain the line $L_1$ and is parallel to $L_2$. The distance of the plane $\pi$ from origin is

• A. $\frac{1}{7}$
• B. $\sqrt{\frac{2}{7}}$
• C. $\sqrt{6}$
• D. None of these

Answer 1

The correct option is B $\sqrt{\frac{2}{7}}$

Equation of plane containing $L_1$ is given by,
$a(x-2)+b(y-1)+c(z+1)=0$
Now this plane is perpendicular to both the given lines
$\Rightarrow a+2c=0..\left(1\right)$
and $a+b-c=0..\left(2\right)$
Solving (1) and (2), we get $a=-2c$ and $b=3c$
Therefore plane is $-2(x-2)+3(y-1)+(z+1)=0$
$\Rightarrow -2x+3y+z+2=0$
Hence, distance of this plane from origin is
$=\left|\frac{2}{2^2+3^2+1^2}\right|=\sqrt{\frac{2}{7}}$