Let $P(2,3,4), A(3,-2,2)$ and $B(6,-17,-4)$.
Let $P$ divides $A B$ in the ratio $k: 1$, then
$2=\frac{6\, k+3}{k+1} $
$\Rightarrow 2\, k+2=6 k+3$
$\Rightarrow -4\, k=1$
$\Rightarrow k=-\frac{1}{4}$
Harmonic conjugate $Q$ divides in the ratio $-k: 1$, i.e. $\frac{1}{4}: 1$
$ Q =\left[\frac{\frac{1}{4}(6)+3}{\frac{1}{4}+1}, \frac{\frac{1}{4}(-17)-2}{\frac{1}{4}+1}, \frac{\frac{1}{4}(-4)+2}{\frac{1}{4}+1}\right] $
$=\left(\frac{6+12}{5}, \frac{-17-8}{5}, \frac{-4-8}{5}\right) $
$=\left(\frac{18}{5},-\frac{25}{5}, \frac{4}{5}\right)$
$ =\left(\frac{18}{5},-5, \frac{4}{5}\right)$