Q. Let $L_{1}$ be the line ${\overrightarrow{r}}_{1} = 2\widehat{i} + \widehat{j} - \widehat{k} + \lambda\left( \widehat{i} + 2\widehat{k} \right)$ and let $L_{2}$ be the line ${\overrightarrow{r}}_{2} = 3\widehat{i} + \widehat{j} + \mu(\widehat{i} + \widehat{j} - \widehat{k})$. Let $\pi$ be the plane which contains the line $L_{1}$ and is parallel to $L_{2}$. The distance of the plane $\pi$ from the origin is
A) $\sqrt{2/7}$
B) $1/7$
C) $\sqrt{6}$
D) None
Explanation
(a)
Equation of the plane containing $L_{1},A(x - 2) + B(y - 1) + C\left( \mathcal{z} + 1 \right) = 0$
Where $A + 2C = 0;A + B - C = 0$
$\Rightarrow A = - 2C,\ B = 3C,\ C = C$
$\Rightarrow$ Plane is $- 2(x - 2) + 3(y - 1) + \mathcal{z} + 1 = 0$ or $2x - 3y - \mathcal{z} - 2 = 0$
Hence, $p = \left| \frac{- 2}{\sqrt{14}} \right| = \sqrt{\frac{2}{7}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.