Given that |z−1|=1, where z is a non zero point on the complex plane, then z−2z 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 Aitan(argz) Let z=r(cosθ+isinθ) Also, |z−1|=1 ⇒|r(cosθ+isinθ)−1|=1 ⇒√(rcosθ−1)2+r2sin2θ=1 ⇒(rcosθ−1)2+r2sin2θ=1 ⇒r2−2rcosθ=0 ⇒r=2cosθ and z¯¯¯z=|z|2⇒1z=¯¯¯z|z|2 Now z−2z=1−2z=1−2¯¯¯z|z|2 =1−2⋅2cosθ(cosθ−isinθ)4cos2θ=1−1+itanθ=itanθ=itan(arg(z))