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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 B itan(argz)
Let z=r(cosθ+isinθ)
Also, |z1|=1
 |r(cosθ+isinθ)1|=1
 (rcosθ1)2+r2sin2θ=1
 (rcosθ1)2+r2sin2θ=1
 r22rcosθ=0
 r=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))

Mathematics

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