Let z be non real number such that (1+z+z2/1-z+z2) ∈ R, then value of 7|z| is

Question

Let z be non real number such that (1+z+z2/1-z+z2) ∈ R, then value of 7|z| is (A) 1 (B) 3 (C) 5 (D) 7

Tardigrade Admin · Accepted Answer

(1+z+z2/1-z+z2)=1+(2 z/1-z+z2) is real ⇒ 1+(2/z+(1)z-1) is real ⇒ z+(1/z) is real ⇒ z+(1/z)= barz+(1/ barz) ⇒ (z- barz)=(1/ barz)-(1/ barz) ⇒ (z- barz)(1-(1/|z|2))=0 ⇒ |z|=1

Q. Let z be non real number such that 1−z+z21+z+z2​∈R, then value of 7∣z∣ is

1

3

5

7

1−z+z21+z+z2​=1+1−z+z22z​ is real ⇒1+z+z1​−12​ is real ⇒z+z1​ is real ⇒z+z1​=zˉ+zˉ1​ ⇒(z−zˉ)=zˉ1​−zˉ1​ ⇒(z−zˉ)(1−∣z∣21​)=0 ⇒∣z∣=1

Q. Let $z$ be non real number such that $\frac{1 + z + z ^{2}}{1 - z + z ^{2}} \in R$ , then value of $7∣ z ∣$ is