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 $\overrightarrow{OP}$ on the axes are $\frac{13}{5},\frac{19}{5},\frac{26}{5}$ respectively, then $P$ divides $QR$ in ratio

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

Answer 1

The correct option is C $3:2$

Since $\overrightarrow{OP}$ has projection $\frac{13}{5},\frac{19}{5}$ and $\frac{26}{5}$ on the coordinate axes, therefore $\overrightarrow{OP}=\frac{13}{5}i+\frac{19}{5}j+\frac{26}{5}k$

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

The position vector of $P$ is $\left(\frac{3\lambda +2}{\lambda +1}\right)i+\left(\frac{5\lambda +2}{\lambda +1}\right)j+\left(\frac{6\lambda +4}{\lambda +1}\right)k$

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

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

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