If z1 and z2 both satisfy the relation z + barz = 2|z - 1| and arg (z1 - z2) = π/4, then Im(z1 + z2) equals

Question

If z1 and z2 both satisfy the relation z + barz = 2|z - 1| and arg (z1 - z2) = π/4, then Im(z1 + z2) equals (A) 0 (B) 1 (C) 2 (D) 3

Tardigrade Admin · Accepted Answer

Let z = a + ib ∴ barz = a - ib ⇒ a = ( z + barz/2) = Real part of z ⇒ 2a = 2|a -1 + ib| (∵ z + barz = 2| z - 1|) ⇒ 2a = 1 + b2 ⇒ 2(a1 - a2) = b12 - b22 = (b1 + b2)(b1 - b2) ...(i) But arg(z1 - z2) = π/4 ∴ tan-1((b1 - b2/a1 - a2)) = π/4 ⇒ b1 - b2 = a1 - a2 ...(ii) Using (ii) in (i) we get, b1 + b2 = 2 ⇒ Im (z 1 + z2) = 2

Q. If $z_{1}$ and $z_{2}$ both satisfy the relation $z + \overset{z}{ˉ} = 2∣ z - 1∣$ and $a r g (z_{1} - z_{2}) = π /4$ , then $I m (z_{1} + z_{2})$ equals

$0$

$1$

$2$

$3$

Q. If z1​ and z2​ both satisfy the relation z+zˉ=2∣z−1∣ and arg(z1​−z2​)=π/4, then Im(z1​+z2​) equals

0

1

2

3

Q. If $z_{1}$ and $z_{2}$ both satisfy the relation $z + \overset{z}{ˉ} = 2∣ z - 1∣$ and $a r g (z_{1} - z_{2}) = π /4$ , then $I m (z_{1} + z_{2})$ equals

$0$

$1$

$2$

$3$