question_answer

If the point \[{{z}_{1}}=1+i\] where \[i=\sqrt{-1}\] is the reflection of a point \[{{z}_{2}}=x+iy\] in the line \[i\bar{z}-iz=5,\] then the point \[{{z}_{2}}\] is     

A) \[1+4i\]

B) \[4+i\]

C) \[1-i\]    

D) \[-1-i\]

Correct Answer: A

Solution :

\[\operatorname{Let}\, z=a+bi\] \[\Rightarrow \,\,\,\bar{z} = a\,- bi\] \[\therefore \,\,\,\,i\bar{z}-iz=i[(a\,-bi)-(a+bi)]1=5\] \[\Rightarrow \,\,\,\,i\left[ -\,2bi \right]=5\] \[\Rightarrow \,\,\,\,b=\frac{5}{2}\] So from figure it is clear that \[\operatorname{x}=1,\,\,\,y=\,\frac{5}{2}+\frac{3}{2}=4\] \[{{z}_{2}}=1+4\,i\]