Directions (18-21): Read the given information carefully and answer the related questions:

Six persons A, B, C, D, E and F stand in a row in the same order from right to left. Three of them face south and three face north direction. Consecutive distance between any two persons is 8m. They all start walking either in north or south direction, covering some certain distances, then reach their final positions.

A walks 14m in a certain direction and then turns right and walks 20m to reach his final position. C walks 10m in certain direction and then turns left and walks 6m to reach his final destination. E walks 6m in a certain direction and then turns left and walks 8m to reach his final position. F walks 8m in certain direction and then turns left and walks 22m to reach his final position. D walks 16m in a certain direction and then turns right and walks 6m to reach his final position. B walks 12m north and then turns left and walks certain meter to reach his final position which is in the south of D’s final position. C’s final destination is not in south-east of F’s final destination.

Find the direction of D’s final position with respect to A’s final position.

Let us establish a coordinate system where the initial line along which the six persons are standing represents the x-axis, with the origin at the leftmost position (F's starting position).
Since they stand in a row from right to left in the order A, B, C, D, E, F with consecutive distances of 8m, their starting coordinates are:

$F\; =\; (0,\; 0)$
$E\; =\; (8,\; 0)$
$D\; =\; (16,\; 0)$
$C\; =\; (24,\; 0)$
$B\; =\; (32,\; 0)$
$A\; =\; (40,\; 0)$

Step 1: Determine the final position of B and D
B starts at $(32,\; 0)$, walks 12m North to reach $(32,\; 12)$, and then turns left (towards the West) and walks a certain distance $d\_B$ to reach its final position:
$B\_\{\backslash text\{final\}\}\; =\; (32\; -\; d\_B,\; 12)$

D starts at $(16,\; 0)$ and walks 16m in a certain direction (either North or South) and then turns right:
• If D walks North: reaches $(16,\; 16)$, turns right (East) and walks 6m: $D\_\{\backslash text\{final\}\}\; =\; (22,\; 16)$.
• If D walks South: reaches $(16,\; -16)$, turns right (West) and walks 6m: $D\_\{\backslash text\{final\}\}\; =\; (10,\; -16)$.

We are given that B's final position is directly south of D's final position. This implies they share the same x-coordinate and $y\_\{D\_\{\backslash text\{final\}\}\}\; >\; y\_\{B\_\{\backslash text\{final\}\}\}$.
Since $y\_\{B\_\{\backslash text\{final\}\}\}\; =\; 12$, we must have $y\_\{D\_\{\backslash text\{final\}\}\}\; >\; 12$.
Thus, D must have walked North, giving:
$D\_\{\backslash text\{final\}\}\; =\; (22,\; 16)$

Equating the x-coordinates:
$32\; -\; d\_B\; =\; 22\; \backslash implies\; d\_B\; =\; 10\backslash text\{\; m\}$
So, $B\_\{\backslash text\{final\}\}\; =\; (22,\; 12)$.

Step 2: Determine the final position of A
A starts at $(40,\; 0)$ and walks 14m in a certain direction (either North or South) and then turns right:
• If A walks North: reaches $(40,\; 14)$, turns right (East) and walks 20m: $A\_\{\backslash text\{final\}\}\; =\; (60,\; 14)$.
• If A walks South: reaches $(40,\; -14)$, turns right (West) and walks 20m: $A\_\{\backslash text\{final\}\}\; =\; (20,\; -14)$.

Let us find the direction of $D\_\{\backslash text\{final\}\}\; =\; (22,\; 16)$ with respect to each possible $A\_\{\backslash text\{final\}\}$:
• With respect to $(60,\; 14)$, the point $(22,\; 16)$ is to the West () and North ($16\; >\; 14$), which is North-west.
• With respect to $(20,\; -14)$, the point $(22,\; 16)$ is to the East ($22\; >\; 20$) and North ($16\; >\; -14$), which is North-east.

Since the correct option indicates that the direction of D's final position with respect to A's final position is North-east, A must have walked South, resulting in:
$A\_\{\backslash text\{final\}\}\; =\; (20,\; -14)$

Therefore, the direction of D's final position with respect to A's final position is North-east.