question_answer

The length of perpendicular from (1, 6, 3) to  the\[\text{line}\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\]is:

A)  3                            

B)         \[\sqrt{11}\]     

C)         \[\sqrt{13}\]                     

D)  5

Correct Answer: C

Solution :

Given, line \[\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\] \[\Rightarrow \]               \[\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}=\lambda \]        (say)                 \[\Rightarrow \]               \[x=\lambda ,y=2\lambda +1,z=3y+2\] Therefore, direction ratios of PQ are                 \[\lambda -1,2\lambda +1-6,3\lambda +2-3\]                 \[\Rightarrow \]               \[\lambda -1,2\lambda -5,3\lambda -1\] \[\because \] PQ is perpendicular to the given line. Therefore,                 \[1(\lambda -1)+2(2\lambda -5)+3(3\lambda -1)=0\]                 \[\Rightarrow \]               \[\lambda =1\] \[\therefore \] The coordinate of Q (1, 3, 5). \[\therefore \]Length of perpendicular                                 \[=\sqrt{{{(1-1)}^{2}}+{{(3-6)}^{2}}+{{(5-3)}^{2}}}\]                                 \[=\sqrt{9+4}=\sqrt{13}.\]