Question

If the points $P(3,2,-4),Q(9,8,-10)$ and $R(5,4,-6)$ are collinear, then $R$ divides $PQ$ in the ratio

• A. $-1:2$
• B. $1:2$
• C. $2:1$
• D. $-2:1$

Answer 1

The correct option is B $1:2$

Given $P(3,2,-4)$

$Q(9,8,-10)R(5,4,-6)$

are collinear so

$R$ divides $PQ$ externally & internally.

$P(x,y,z)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

so $(x,y,z)=(5,4,-6)$

$(x_1,y_1,z_1)=(3,2,-4)$

$(x_2,y_2,z_2)=(9,8,-10)$

Ratio $k:1=m:n$

So:-

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

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

So :- $\frac{9k+3}{k+1}=5$

$9k+3=5k+5$

$4k=2$

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

$\therefore k:1=1:2$

$\therefore R$ divides $PQ$ in Ratio $1:2$.