If $(a, b)$ is the mid-point of the line segment joining the points $A (10, -6)$ and $B (k, 4)$ and $a - 2b = 18$,find the value of $k$ and the distance $AB$.

(D) Since $(a, b)$ is the mid-point of the line segment $AB$,we use the mid-point formula:
$(a, b) = \left(\frac{10 + k}{2}, \frac{-6 + 4}{2}\right)$
$(a, b) = \left(\frac{10 + k}{2}, -1\right)$
Equating the coordinates,we get:
$a = \frac{10 + k}{2}$ ... $(i)$
$b = -1$ ... $(ii)$
Given the equation $a - 2b = 18$,substitute $b = -1$:
$a - 2(-1) = 18$
$a + 2 = 18 \Rightarrow a = 16$
Now,substitute $a = 16$ into equation $(i)$:
$16 = \frac{10 + k}{2}$
$32 = 10 + k \Rightarrow k = 22$
Thus,the coordinates of $B$ are $(22, 4)$.
Now,calculate the distance $AB$ using the distance formula:
$AB = \sqrt{(22 - 10)^2 + (4 - (-6))^2}$
$AB = \sqrt{(12)^2 + (10)^2}$
$AB = \sqrt{144 + 100} = \sqrt{244}$
$AB = 2\sqrt{61}$ units.