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Question

Let z1 and z2 be complex numbers such that z1z2 and |z1|=|z2|. If Re(z1)>0 and Im(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
Let z1=a+ib, z2=c+id
|z1|=|z2|
a2+b2=c2+d2
a2+b2c2d2=0 ...(1)

z1+z2z1z2=(a+c)+i(b+d)(ac)+i(bd)
=[(a+c)+i(b+d)(ac)2+(bd)2]×[(ac)i(bd)]
=2i(adbc)(ac)2+(bd)2

z1+z2z1z2 is purely imaginary.

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