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Question

If z1z2 and |z1+z2|=1z1+1z2 then

A
at least one of z1,z2 is unimodular
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B
both z1,z2 are unimodular
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C
z1.z2=1
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D
None of these
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Solution

The correct option is D None of these
Let
z1=x1+iy1
z2=x2+iy2
Now
z1+z2=(x1+x2)+i(y1+y2)
1z1+1z2
=¯z1+¯z2
=(x1+x2)i(y1+y2).
Now
|z1+z2|
=(x2+x1)2+(y2+y1)2
And
|1z1+1z2|
=|(x1+x2)i(y1+y2)|
=(x2+x1)2+(y2+y1)2
Hence
|z1+z2|=|1z1+1z2| for all values of z1 and z2.

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