InterviewSolution
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InterviewSolution
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Acomplex number z is said to be unimodular if . Suppose `z_1`and `z_2`are complex numbers such that `(z_1-2z_2)/(2-z_1z_2)`is unimodular and `z_2`is not unimodular. Then the point `z_1`lieson a :(1)straight line parallel to x-axis(2) straight line parallel to y-axis(3)circle of radius 2(4) circle of radius `sqrt(2)`A. straight line parallel to x-axisB. straight line parallel to y-axiesC. circle of radius 2D. circle of radius `sqrt(2)` |
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Answer» Correct Answer - C If is unimodular then |z|=1 also use property of modules I,e `zz=|z|^2` Given `z_2` is not unimodular I,e `|z_2|ne 1` and `(z_1-2z_2)/(2-z_1barz_2)|` is unimodular `rArr |(z_1-2z_2)/(2-z_1barz_2)|=1 rArr |z_1-2z_2|^2=|2-z_1barz_2|^2` `rArr ( z_1-2z_2)(barz_1-2barz_2)=(2-z_1bar z_2)(2-barz_1z_2)" "[zbarz=|z|^2]` `rArr |z|^2+4|^2-2barz_1z_2-2z_1bar z_2` `rArr 4+|z_1|^2|z_2|^2-2barz_1z_2-2z_1barz_2rArr(|z_2 |^2-1)(|z_1|^2-4)=0` `because " "|z_2|ne1` `therefore |z_1|=2` Let `z_1=x+iy rArr x^2+y^2=(2)^2` `therefore`Point `z_1`lies on circle of radius |
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