Find the ratio in which the point

Let the ratio be k:1 in which point R divides point P and point Q.

Using $\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$, we get,

Here $m$ and $n$ are $k$ and 1 . The point which this formula gives is already given, i.e. $\mathrm{R}(5,4,-6)$ and the joining points are $\mathrm{P}(3,2,-4)$ and $\mathrm{Q}(9,8,-10)$.

$\left.(5,4,-6)=\frac{\mathrm{k} \times 9+1 \times 3}{\mathrm{k}+1}, \frac{\mathrm{k} \times 8+1 \times 2}{\mathrm{k}+1}, \frac{\mathrm{k} \times-10+1 \times-4}{\mathrm{k}+1}\right)$

Taking any point and finding the value of k, we get

$5=\frac{\mathrm{k} \times 9+1 \times 3}{\mathrm{k}+1}$

$5 k+5=9 k+3$

$4 k=2$

$\mathrm{K}=\frac{1}{2}$

Therefore, the ratio be $1: 2$.