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Question

Suppose z and ω are two complex numbers such that |z|1,|ω|1, and |z+iω|=|ziω|=2.

Which of the following is true for z and ω?

A
Re(z)=Re(ω)
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B
Im(z)=Im(ω)
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C
Re(z)=Im(ω)
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D
Im(z)=Re(ω)
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Solution

The correct option is D Im(z)=Re(ω)
Suppose z=x+iy,ω=α+iβ

|z+iω|=2

|z|2+|ω|2+iω¯¯¯zi¯¯¯ωz=4

iω¯¯¯z=i¯¯¯ωz=2 ......... (1)

Also |zi¯¯¯ω|=2

|z|2+|ω|2+iωzi¯¯¯ω¯¯¯z=4

iωzi¯¯¯ω¯¯¯z=2

Adding (1) and (2), we get
i(ω+¯¯¯ω)(z¯¯¯z)=4

i(2iβ)(2x)=4

βx=1....... (3)

Subtracting (1) and (2), we get

i(ω+¯¯¯ω)(z¯¯¯z)=0αy=0........ (4)

either α=0 or y=0

If y=0, then x2+y2=1

x=±1 and z=1 or 1

If α=0 the $\alpha^2+\beta^2=1\Rightarrow \beta =\pm 1\Rightarrow \omega =\pm i$

So Im(z)=Re(ω)=0

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