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 $\overrightarrow{OP}$ 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

The correct option is B 3 : 2

Let $P(x,y,z)$

$Q(2,2,4)$

$R(3,5,6)$

Let the ratio in which $P(x,y,z)$ divides the line joining $Q(2,2,4),R(3,5,6)$ be $\lambda :1$

According to the section formula ,

$P(x,y,z)=(\frac{3\lambda +2}{\lambda +1},\frac{5\lambda +2}{\lambda +1},\frac{6\lambda +4}{\lambda +1})$

Projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ is $\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{\left|\overrightarrow{b}\right|}$

It is given that projection of $\overrightarrow{OP}$ on coordinate axes are $\frac{13}{5},\frac{19}{5},\frac{26}{5}$ is

$\frac{(x,y,z)\cdot (1,0,0)}{|1,0,0|}=\frac{13}{5}$

$x=\frac{13}{5}=\frac{3\lambda +2}{\lambda +1}$

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