question_answer

If the position vectors of two points P and Q are respectively \[9\,\hat{i}-\hat{j}+5\hat{k}\]and \[\hat{i}+3\hat{j}+5\hat{k}\] and the line segment PQ intersects the YOZ plane at a point R, then \[PR:RQ\]is equal to

A)  \[9:1\]         

B)  \[1:9\]

C)  \[-1:9\]

D)  \[-9:1\]

Correct Answer: D

Solution :

Position vector of \[P=9\hat{i}-\hat{j}+5\hat{k}=(9,-1,5)\] and position vector of Q \[=\hat{i}+3\hat{j}+5\hat{k}=(1,3,5)\] Let \[PR:RQ=m:1\] \[\because \] PQ intersects the YOZ plane. \[\therefore \] x-coordinate of \[R=0\] \[\Rightarrow \] \[\frac{9\times 1+m\times 1}{m+1}=0\] \[\Rightarrow \] \[9+m=0\] \[\Rightarrow \] \[m=-9\] \[\therefore \] Ratio \[=-9:1\]