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Question

For any two complex numbers z1,z2 we have |z1+z2|2=|z1|2+|z2|2. Then

A
Re(z1z2)=0
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B
Im(z1z2)=0
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C
Re(z1z2)=0
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D
Im(z1z2)=0
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Solution

The correct option is A Re(z1z2)=0

We know that,
|z1+z2|2=|z1|2+|z2|2+2Re(z1¯¯¯z2)(i)
Given,
|z1+z2|2=|z1|2+|z2|2(ii)
From equation (i) and (ii), we get
2Re(z1¯¯¯¯¯z2)=0
z1¯¯¯¯¯z2+¯¯¯¯¯¯¯¯¯z1¯¯¯¯¯z2=0 (2Re(z)=z+¯¯¯z,¯¯¯¯¯¯z=z)
z1¯¯¯¯¯z2+¯¯¯¯¯z1z2=0
z1¯¯¯¯¯z2=¯¯¯¯¯z1z2z1z2=¯¯¯¯¯z1¯¯¯¯¯z2z1z2+¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z1z2)=0(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z1z2)=¯¯¯¯¯z1¯¯¯¯¯z2)
2Re(z1z2)=0
Re(z1z2)=0

Alternate Solution
We know that,
|z1+z2|2=|z1|2+|z2|2+2Re(z1¯¯¯z2)(i)
Given,
|z1+z2|2=|z1|2+|z2|2(ii)
From equation (i) and (ii), we get
2Re(z1¯¯¯¯¯z2)=0
Re(z1z2¯¯¯¯¯z2z2)=0
[z¯¯¯z=|z|2]
Re(z1z2|z2|2)=0
z1z2|z2|2 is purely imaginary
So, z1z2 will also be purely imaginary
Re(z1z2)=0


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