Given point $R$ lies on the line segment joining the points $P$ and $Q,$ hence direction ratios of the line
$PR$ and $PQ$ must be identical

$\Rightarrow \frac{2}{6}=\frac{y+3}{3}=\frac{z-4}{6}$
$\Rightarrow y=-2$ and $z=6$
$\Rightarrow R\left(4, -2, 6\right).$
Hence, distance from origin $=\sqrt{16+4+36}=2\sqrt{14}$