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# Given that $P(3,2,-4),\:Q(5,4,-6)\:\:and\:\:R(9,8,-10)$ are collinear. Find the ratio in which $Q$ divides $PR$

Toolbox:
• Section formula: The coordinates of the point $C$ that divides the segment joining the points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ in the ratio $l:m$ internally is given by $C\big(\large\frac{lx_2+mx_1}{l+m},\frac{ly_2+my_1}{l+m},\frac{lz_2+mz_1}{l+m}\big)$
Given: $P(3,2,-4),\:Q(5,4,-6)\:\:and\:\:R(9,8,-10)$ are collinear.
Let the ratio in which $Q$ divides $PR$ be $k:1$
$\therefore$ We know from section formula that the coordinates of $Q$ is given by
$Q\big(\large\frac{9k+3}{k+1},\frac{8k+2}{k+1},\frac{-10k-4}{k+1}\big)$
But given that $Q(5,4,-6)$
$\Rightarrow\:5=\large\frac{9k+3}{k+1}$
$\Rightarrow\:5k+5=9k+3$
$\Rightarrow\:4k=2$ or $k=\large\frac{1}{2}$