If $P(x, y, z) $ is a point on the line segment joining $Q(2, 2, 4)$ and $R(3, 5, 6)$ such that projections of $\vec{OP}$ on the axes are $\frac{13}{5}, \frac{19}{5}, \frac{26}{5}$ respectively, then P divides QR in the ratio 

3 : 2

The coordinates of P are $\left(\frac{13}{5}, \frac{19}{5}, \frac{26}{5}\right).$

Suppose , P divides QR in the ratio λ : 1. Then, the coordinates of P are $\left(\frac{3λ+2}{λ+1}, \frac{5λ+2}{λ+1},\frac{6λ+4}{λ+1}\right)$

$∴ \frac{3λ+2}{λ+1}=\frac{13}{5}, \frac{5λ+2}{λ+1}=\frac{19}{5}, \frac{6λ+4}{λ+1}=\frac{26}{5}⇒λ =\frac{3}{2}$

Hence, required ratio is 3 : 2.