In $\mathbb { R } ^ { 3 }$, let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P _ { 1 } : x + 2 y - z + 1 = 0$ and $P _ { 2 } : 2 x - y + z - 1 = 0$. Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane $P _ { 1 }$. Which of the following points lie(s) on $M$? (A) $\left( 0 , - \frac { 5 } { 6 } , - \frac { 2 } { 3 } \right)$ (B) $\left( - \frac { 1 } { 6 } , - \frac { 1 } { 3 } , \frac { 1 } { 6 } \right)$ (C) $\left( - \frac { 5 } { 6 } , 0 , \frac { 1 } { 6 } \right)$ (D) $\left( - \frac { 1 } { 3 } , 0 , \frac { 2 } { 3 } \right)$
In $\mathbb { R } ^ { 3 }$, let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P _ { 1 } : x + 2 y - z + 1 = 0$ and $P _ { 2 } : 2 x - y + z - 1 = 0$. Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane $P _ { 1 }$. Which of the following points lie(s) on $M$?\\
(A) $\left( 0 , - \frac { 5 } { 6 } , - \frac { 2 } { 3 } \right)$\\
(B) $\left( - \frac { 1 } { 6 } , - \frac { 1 } { 3 } , \frac { 1 } { 6 } \right)$\\
(C) $\left( - \frac { 5 } { 6 } , 0 , \frac { 1 } { 6 } \right)$\\
(D) $\left( - \frac { 1 } { 3 } , 0 , \frac { 2 } { 3 } \right)$