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 \[PL\,\,(2\lambda ,\,\,-3\lambda -1,\,\,8\lambda -10)\]

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