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Question

Given that |z1|=1, where z is a non zero point on the complex plane, then z2z is equal to :

A
itan(argz)
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B
icot(argz)
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C
cot(argz)
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D
tan(argz)
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Solution

The correct option is A itan(argz)
Let z=r(cosθ+isinθ)
Also, |z1|=1
|r(cosθ+isinθ)1|=1(rcosθ1)2+r2sin2θ=1(rcosθ1)2+r2sin2θ=1r22rcosθ=0r=2cosθ
And
z¯¯¯z=|z|21z=¯¯¯z|z|2
Now,
z2z=12z=12¯¯¯z|z|2=122cosθ(cosθisinθ)4cos2θ=11+itanθ=itanθ=itan(arg(z))

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