Question

If $P(x,y,x)$ 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 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

The correct option is B $3:2$

Since $OP$ has projection $\frac{13}{5},\frac{19}{5}$ and $\frac{26}{5}$ on the coordinate axis,

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

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