Question

The coordinates of the points which trisect the line segments joining the points $P(4,2,-6)$ and $Q(10,-16,6)$, are

• A. $(6,-4,-2)$ and $(8,10,-2)$
• B. $(6,-4,-2)$ and $(8,-10,2)$
• C. $(-6,4,2)$ and $(-8,10,2)$
• D. None of these

Answer 1

Answer: (B)

Let the point $R_1$ and $R_2$ trisects the line $PQ$ i.e., $R_1$ divides the line in the ratio $1:2$.

$\Rightarrow R_1=\left(\frac{1\times 10+2\times 4}{1+2},\frac{1\times (-16)+2\times 2}{1+2},\frac{1\times 6+2\times (-6)}{1+2}\right)$
$=\left(\frac{10+8}{3},\frac{-16+4}{3},\frac{6-12}{3}\right)=\left(\frac{18}{3},\frac{-12}{3},\frac{-6}{3}\right)$
$=(6,-4,-2)$
Again, let the point $R_2$ divides $PQ$ internally in the ratio $2:1$. Then,

$\Rightarrow R_2=\left(\frac{2\times 10+1\times 4}{2+1},\frac{2\times (-16)+1\times 2}{2+1},\frac{2\times 6+1\times (-6)}{1+2}\right)$
$=\left(\frac{20+4}{3},\frac{-32+2}{3},\frac{12-6}{3}\right)=\left(\frac{24}{3},\frac{-30}{3},\frac{6}{3}\right)$
$=(8,-10,2)$
Hence, required points are $(6,-4,-2)$ and $(8,-10,2)$.