Let $\ell _ { 1 }$ and $\ell _ { 2 }$ be the lines $\vec { r } _ { 1 } = \lambda ( \hat { i } + \hat { j } + \hat { k } )$ and $\vec { r } _ { 2 } = ( \hat { j } - \hat { k } ) + \mu ( \hat { i } + \hat { k } )$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell _ { 1 }$. For a plane $H$, let $d ( H )$ denote the smallest possible distance between the points of $\ell _ { 2 }$ and $H$. Let $H _ { 0 }$ be a plane in $X$ for which $d \left( H _ { 0 } \right)$ is the maximum value of $d ( H )$ as $H$ varies over all planes in $X$.

Match each entry in List-I to the correct entries in List-II.

\textbf{List-I}

(P) The value of $d \left( H _ { 0 } \right)$ is

(Q) The distance of the point $( 0,1,2 )$ from $H _ { 0 }$ is

(R) The distance of origin from $H _ { 0 }$ is

(S) The distance of origin from the point of intersection of planes $y = z , x = 1$ and $H _ { 0 }$ is

\textbf{List-II}

(1) $\sqrt { 3 }$

(2) $\frac { 1 } { \sqrt { 3 } }$

(3) 0

(4) $\sqrt { 2 }$

(5) $\frac { 1 } { \sqrt { 2 } }$

The correct option is:

(A) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 4 )$ $( R ) \rightarrow ( 5 )$ $( S ) \rightarrow ( 1 )$

(B) $( P ) \rightarrow ( 5 )$ $( Q ) \rightarrow ( 4 )$ $( R ) \rightarrow ( 3 )$ $( S ) \rightarrow ( 1 )$

(C) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 1 )$ $( R ) \rightarrow ( 3 )$ $( S ) \rightarrow ( 2 )$

(D) $( P ) \rightarrow ( 5 )$ $( Q ) \rightarrow ( 1 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 2 )$