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Question

If w=α+iβ,where β0 and z1, satisfies the condition that (w¯wz1z) is purely real, then the set of values of z is

A
|z|=1,z2
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B
|z|=1, and z1
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C
z=¯z
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D
None of these
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Solution

The correct option is B |z|=1, and z1
Let z1=w¯wz1z be purely realz1=¯z1w¯wz1z=¯ww¯z1¯zww¯z¯wz+¯wz.¯z=¯wz¯ww¯z+wz.¯z(w¯w)+(¯ww)|z|2=0(w¯w)(1|z|2)=0|z|2=1 [as w¯w0, since β0]|z|=1 and z1

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