A) 1 : 2
B) 3 : 2
C) 2 : 3
D) 1 : 3
E) 3 : 1
Correct Answer: B
Solution :
Since,\[\overrightarrow{OP}\]has projections\[\frac{13}{5},\frac{19}{5}\]and\[\frac{26}{5}\]on the coordinate axes, therefore \[\overrightarrow{OP}=\frac{13}{5}\hat{i}+\frac{19}{5}\hat{j}+\frac{26}{5}\hat{k}\]. Suppose P divides the line joining Q(2, 2, 4) and R (3, 5, 6) in the ratio\[\lambda :1\]. Then, the position vector of\[P\]is \[\left( \frac{3\lambda +2}{\lambda +1} \right)\hat{i}+\left( \frac{5\lambda +2}{\lambda +1} \right)\hat{j}+\left( \frac{6\lambda +4}{\lambda +1} \right)\hat{k}\] \[\therefore \] \[\frac{13}{5}i+\frac{19}{5}\hat{j}+\frac{26}{6}\hat{k}=\left( \frac{3\lambda +2}{\lambda +1} \right)\hat{i}\] \[+\left( \frac{5\lambda +2}{\lambda +1} \right)\hat{j}\left( \frac{6\lambda +4}{\lambda +1} \right)\hat{k}\] \[\Rightarrow \] \[\frac{3\lambda +2}{\lambda +1}=\frac{13}{5},\frac{5\lambda +2}{\lambda +1}=\frac{19}{5}\] And \[\frac{6\lambda +4}{\lambda +1}=\frac{26}{5}\] \[\Rightarrow \] \[2\lambda =3\Rightarrow \lambda =\frac{3}{2}\] Hence, P divides QR in the ratio\[3:2\].
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