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Question

Let z1 and z2 be complex numbers such that z1z2 and |z1|=|z2|. If Re(z1)>0 and lm(z2)<0, then z1+z2z1z2 is

A
one
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B
real and positive
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C
real and negative
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D
purely imaginary
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Solution

The correct option is D purely imaginary
z1=x1+iy1;z2=x2+iy2
Re(z1)>0x1>0 and Im(z2)<0y2<0
|z1|=|z2||z1|2=|z2|2
z1¯¯¯¯¯z1=z2¯¯¯¯¯z2
Now (z1+z2z1z2)+(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯z1+z2z1z2)
(z1+z2z1z2)+(¯¯¯¯¯z1+¯¯¯¯¯z2¯¯¯¯¯z1¯¯¯¯¯z2)=z1¯¯¯¯¯z1+z2¯¯¯¯¯z1z2¯¯¯¯¯z2+z1¯¯¯¯¯z2z2¯¯¯¯¯z1z2¯¯¯¯¯z1(z1z2)(¯¯¯¯¯z1¯¯¯¯¯z2)
=2(|z1|2=|z2|2)(z1z2)(¯¯¯¯¯z1¯¯¯¯¯z2)=0(|z1|2=|z2|2)
z1+z2z1z2 is purely imaginary

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