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# 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 coordinate axes are $\large\frac{13}{5},\frac{19}{5},\frac{26}{5}$ respectively, then $P$ divides $QR$ in the ratio ?

$\begin{array}{1 1} (a)\:1:2\:\:\:\qquad\:\:(b)\:\:3:2\:\:\:\qquad\:\:(c)\:\:2:3\:\:\:\qquad\:\:(d)\:\:1:3 \end{array}$

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• Projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow b|}$
Let the ratio in which $P(x,y,z)$ divides the line joining $Q(2,2,6)$ and $R(3,5,6)$ be $\lambda:1$
$\therefore$ According to section formula $P$ is given by $(\large\frac{3\lambda+2}{\lambda+1},\frac{5\lambda+2}{\lambda+1},\frac{6\lambda+6}{\lambda+1})$
Also it is given that projection of $\overrightarrow {OP}$ on coordinate axes are $\large\frac{13}{5},\frac{19}{5},\frac{26}{5}$
$\Rightarrow\:\large\frac{(x,y,z).(1,0,0)}{|(1,0,0)|}=\frac{13}{5}$
$\large\frac{(x,y,z).(0,1,0)}{|(0,1,0)|}=\frac{19}{5}$ and $\large\frac{(x,y,z).(0,0,1)}{|(0,0,1)|}=\frac{26}{5}$
$\Rightarrow\:x=\large\frac{13}{5},$$y=\large\frac{19}{5}$ and $z=\large\frac{26}{5}$
$P(x,y,z)=(\large\frac{3\lambda+2}{\lambda+1},\frac{5\lambda+2}{\lambda+1},\frac{6\lambda+6}{\lambda+1})$
$\Rightarrow\:\large\frac{13}{5}=\frac{3\lambda+2}{\lambda+1}$
$\Rightarrow\:\lambda=\large\frac{3}{2}$
$\therefore$ the ratio is $3:2$