Find the equation of a plane which bisects perpendicularly the line joining the points $A(2,3,4)$ and $B(4,5,8)$ at right angles.

Since, the equation of a plane is bisecting perpendicular the line joining the points

$A(2,3,4)$ and $B(4,5,8)$ at right angles.

So, mid-point of AB is $\left( {\frac{{2 + 4}}{2},\frac{{3 + 5}}{2},\frac{{4 + 8}}{2}} \right)$ i.e., (3,4,6) .

Also, $\overrightarrow {\rm{N}} = (4 - 2)\widehat {\rm{i}} + (5 - 3)\widehat {\rm{j}} + (8 - 4)\widehat {\rm{k}} = 2\widehat {\rm{i}} + 2\widehat {\rm{j}} + 4\widehat {\rm{k}}$

So, the required equation of the plane is $(\overrightarrow {\rm{r}} - \overrightarrow {\rm{a}} ) \cdot \overrightarrow {\rm{N}} = 0$.

$\Rightarrow$ $[(x - 3)\widehat {\rm{i}} + (y - 4)\widehat {\rm{j}} + (z - 6)\widehat {\rm{k}}] \cdot (2\widehat {\rm{i}} + 2\widehat {\rm{j}} + 4\widehat {\rm{k}}) = 0$

$\Rightarrow$ $2x - 6 + 2y - 8 + 4z - 24 = 0$

$\Rightarrow$ $2x + 2y + 4z = 38$
$\therefore$ $x + y + 2z = 19$