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 projection of $\overline{OP}$ on the axes 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 $OP$ has projections $\frac{13}{5},\frac{19}{5}$ and $\frac{26}{5}$ on the coordinate axes,

therefore $OP=\frac{13}{5}\hat{i}+\frac{19}{5}\hat{j}+\frac{26}{5}\hat{k}$

Suppose $P$ divides the join of $Q(2,2,4)$ and $R(3,5,6)$ in the ratio $\lambda :1$

Then, the position vector of $P$ is

$\left(\frac{3\lambda +2}{\lambda +1}\right)\hat{i}+\left(\frac{5\lambda +2}{\lambda +1}\right)\hat{j}+\left(\frac{6\lambda +4}{\lambda +1}\right)\hat{k}$

$\therefore \frac{13}{5}\hat{i}+\frac{19}{5}\hat{j}+\frac{26}{5}\hat{k}=\left(\frac{3\lambda +2}{\lambda +1}\right)\hat{i}+\left(\frac{5\lambda +2}{\lambda +1}\right)\hat{j}+\left(\frac{6\lambda +4}{\lambda +1}\right)\hat{k}$

$\Rightarrow \frac{3\lambda +2}{\lambda +1}=\frac{13}{5},\frac{5\lambda +2}{\lambda +1}=\frac{19}{5},\frac{6\lambda +4}{\lambda +1}=\frac{26}{5}$

$\Rightarrow 2\lambda =3$

$\Rightarrow \lambda =\frac{3}{2}$