z is a complex number such that |z|=1 and w=z−1z+1 then Re(w)=

w=z−1z+1Applying C & Dw+1w−1=2z−2 ⇒z=1+w1−w |z|=1⇒|1+w|=|1−w| ⇒1+|w|2+2Re(w)=1+|w|2−2Re(w) ⇒4Re(w)=0 ⇒Re(w)=0

z is a complex number such that |z|=1 and w=z−1z+1 then Re(w)=

w=z−1z+1Applying C & Dw+1w−1=2z−2 ⇒z=1+w1−w |z|=1⇒|1+w|=|1−w| ⇒1+|w|2+2Re(w)=1+|w|2−2Re(w) ⇒4Re(w)=0 ⇒Re(w)=0

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z is a complex number such that $| z | = 1$ and $w = \frac{z - 1}{z + 1}$ then $Re (w) =$

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$0$

$\frac{- 1}{{| z + 1 |}^{2}}$

$\frac{\sqrt{2}}{{| z + 1 |}^{2}}$

$\frac{1}{{| z + 1 |}^{2}}$

answer is D.

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Detailed Solution

$w = \frac{z - 1}{z + 1}$

Applying C & D

$\frac{w + 1}{w - 1} = \frac{2 z}{- 2} \Rightarrow z = \frac{1 + w}{1 - w} | z | = 1 \Rightarrow | 1 + w | = | 1 - w | \Rightarrow 1 + {| w |}^{2} + 2 Re (w) = 1 + {| w |}^{2} - 2 Re (w) \Rightarrow 4 Re (w) = 0 \Rightarrow Re (w) = 0$

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z is a complex number such that $| z | = 1$ and $w = \frac{z - 1}{z + 1}$ then $Re (w) =$