question_answer

The reflection of the point P (1, 0, 0) in the line \[\frac{x-1}{2}=\frac{y+1}{-\,3}=\frac{z+10}{8}\] is

A) \[(3,\,\,-4,\,\,-2)\]            

B) \[(5,\,\,-8,\,\,-4)\]

C) \[(1,\,\,-1,\,\,-10)\]                      

D) \[(2,\,\,-3,\,\,8)\]

Correct Answer: B

Solution :

\[\frac{x-1}{2}=\frac{y+1}{-\,3}=\frac{z+10}{8}=\lambda \]

\[L\,(2\lambda +1,\,\,-3\lambda -1,\,\,8\lambda -10)\] direction ratio of \[PL\,\,\left\langle 2\lambda ,\,\,-3\lambda -1,\,\,8\lambda -10 \right\rangle \]

PL and AB are perpendicular lines \[2\,(2\lambda )\,-3\,(-3\lambda -1)+8\,(8\lambda -10)=0\]

\[\Rightarrow \]   \[77\lambda -77=0\]       \[\Rightarrow \]   \[\lambda =1\]

\[L\,(3,\,\,-4,\,\,-2)\]

L is the mid point of PQ

\[Q\,({{x}_{1}},\,\,{{y}_{1}},\,\,{{z}_{1}})\]

Then     \[\frac{{{x}_{1}}+1}{2}=3\]    \[\Rightarrow \]   \[{{x}_{1}}=5\]

\[\frac{{{y}_{1}}+0}{2}=-\,4\] \[\Rightarrow \]   \[{{y}_{1}}=-\,8\]

and       \[\frac{{{z}_{1}}+0}{2}=-\,2\]  \[\Rightarrow \]   \[{{z}_{1}}=-\,4\]

reflection point of P is \[(5,\,\,-\,8,\,\,-4)\]