A) 1 : 2
B) 3 : 2
C) 2 : 3
D) 1 : 3
Correct Answer: B
Solution :
Since, \[\overrightarrow{\text{OP}}\]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 the points \[Q(2,2,4)\]and\[R(3,5,6)\]in the ratio \[\lambda :1,\] then position vector of P is \[\left( \frac{3\lambda +2}{\lambda +1} \right)\hat{i}+\left( \frac{5\lambda +2}{\lambda +1} \right)\hat{j}+\left( \frac{6\lambda +4}{\lambda +1} \right)\hat{k}\] \[\therefore \] \[\frac{13}{5}\hat{i}+\frac{19}{5}\hat{j}+\frac{26}{5}\hat{k}\] \[=\left( \frac{3\lambda +2}{\lambda +1} \right)\hat{i}+\left( \frac{5\lambda +2}{\lambda +1} \right)\hat{j}+\left( \frac{6\lambda +4}{\lambda +1} \right)\hat{k}\] \[\Rightarrow \]\[\frac{3\lambda +2}{\lambda +1}=\frac{13}{5},\](comparing the coefficients of\[\hat{i}\]) \[\Rightarrow \]\[15\lambda +10=13\lambda +13\] \[\Rightarrow \]\[2\lambda =3\] \[\Rightarrow \]\[\lambda =\frac{3}{2}\] Hence, P divides QR in the ratio 3 : 2.
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