Question

If $P(x,y,z)$ is a point on the line segment joining $Q(2,2,4)$ and $R(3,5,6)$ such that the projections of $OP$ on the axis are $\frac{13}{5},\frac{19}{5},\frac{26}{5}$ respectively, then $P$ divides $QR$ in the ratio

• A. $1:2$
• B. $3:2$
• C. $2:3$
• D. $1:3$

Answer 1

Answer: (B)

$3:2$

Since, $\overline{OP}$ has projections $\frac{13}{5},\frac{19}{5},\frac{26}{5}$ on the coordinate axes
$\therefore \overline{OP}=\frac{13}{5}\hat{i}+\frac{19}{5}\hat{j}+\frac{26}{5}\hat{k}$
Suppose $P$ divides the line segment joining of $Q(2,2,4)$ and $R(3,5,6)$ in the ratio $\lambda :1$
then the position vector of $P$ is
$\left(\frac{3\lambda +2}{\lambda +1}\right)\hat{i}+\left(\frac{5\lambda +2}{\lambda +1}\right)\hat{j}+\left(\frac{6\lambda +4}{\lambda +1}\right)\hat{k}$
$\therefore \frac{13}{5}\hat{i}+\frac{19}{5}\hat{j}+\frac{26}{5}\hat{k}=\left(\frac{3\lambda +2}{\lambda +1}\right)\hat{i}+\left(\frac{5\lambda +2}{\lambda +1}\right)\hat{j}+\left(\frac{6\lambda +4}{\lambda +1}\right)\hat{k}$
$\Rightarrow \lambda =3/2$
$\therefore$ Required ratio in which $P$ divides $QR$ is $3:2$