Question

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.

Answer 1

Let $Q(5,4,-6)$ divides the line segent joining points $P(3,2,-4)$ and $R(9,8,-10)$ in the ratio k : 1 internally.

$\therefore$ Then coordinates of Q are

$(\frac{9k+3}{k+1},\frac{8k+2}{k+1},\frac{-10k-4}{k+1})$

But it is given that coordinates of Q is $(5,4,-6)$

$\therefore \frac{9k+3}{k+1}=5\Rightarrow 9k+3=5k+5$

$\Rightarrow 4k=2\Rightarrow k=\frac{1}{2}$

Thus Q divides the line segment joining points P and R in the ratio $\frac{1}{2}i.e.1:2$ internally.