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Question

If z1,z2 are two complex numbers such that |z1|=|z2|=2 and argz1+argz2=0, then z1z2=

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Solution

Let z1=|z1|(cosθ+isinθ)
Where θ=arg(z1)
z2=|z2|(cosϕ+isinϕ)
Where ϕ=arg(z2)
It is given that argz2=argz1ϕ=θ
Also,
|z1|=|z2|z2=|z1|[cos(θ)+isin(θ)]z2=|z1|(cosθisinθ)z2=¯¯¯¯¯z1z1=¯¯¯¯¯z2z1z2=¯¯¯¯¯z2z2=|z2|2=4

Alternate Solution :
argz1+argz2=0arg(z1z2)=0z1z2=k, (k>0)
Now,
|z1z2|=kk=|z1||z2|=4

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