Let z1 and z2 be two complex numbers such that |z2|=1 and (z1-1/z1+1)=((z2-1/z2+1))2, then find the value of | z 1-1|+| z 1+1|.

Question

Let z1 and z2 be two complex numbers such that |z2|=1 and (z1-1/z1+1)=((z2-1/z2+1))2, then find the value of | z 1-1|+| z 1+1|. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

∵| z 2|=1 ⇒ z 2= e i θ= cos θ+ i sin θ ∴ ( z 2-1/ z 2+1)=( cos θ-1+ i sin θ/ cos θ+1+ i sin θ)=(-2 sin 2 (θ/2)+ i ⋅ 2 sin (θ/2) cos (θ/2)/2 cos 2 (θ)2+ i ⋅ 2 sin (θ)2 cos (θ/2)=( i sin (θ/2)/ cos (θ)2)= i tan (θ/2) ∴ ( z 1-1/ z 1+1)=- tan 2 (θ/2) ⇒ z 1-1=- z 1 tan 2 (θ/2)- tan 2 (θ/2) ⇒ z 1=(1- tan 2 (θ/2)/1+ tan 2 (θ)2)= cos θ which is purely real and lies in [-1,1] ∴| z 1-1|+| z 1+1|=2

Q. Let z1​ and z2​ be two complex numbers such that ∣z2​∣=1 and z1​+1z1​−1​=(z2​+1z2​−1​)2, then find the value of ∣z1​−1∣+∣z1​+1∣.

Q. Let $z_{1}$ and $z_{2}$ be two complex numbers such that $∣ z_{2} ∣ = 1$ and $\frac{z _{1} - 1}{z _{1} + 1} = (\frac{z _{2} - 1}{z _{2} + 1})^{2}$ , then find the value of $∣ z_{1} - 1 ∣ + ∣ z_{1} + 1 ∣$ .