For any complex numbers z1 and z2, the maximum value of (z1 barz2+ barz1 z2/|z1||z2|) is

Question

For any complex numbers z1 and z2, the maximum value of (z1 barz2+ barz1 z2/|z1||z2|) is (A) 1 / 2 (B) 1 (C) 3 / 2 (D) 2

Tardigrade Admin · Accepted Answer

(z1 barz2+ barz1 z2/|z1||z2|) =2 textRe((z1 barz2/|z1 z2|)) ≤ 2 ⋅ (|z1 barz2|/|z1 z2|)=2 Alternatively, z1=r1 ei α and z2=r2 ei β ∴ (z1 barz2+ barz1 z2/|z1 | z2|) =ei(α-β)+e-i(α-β) =2 cos (α-β) ≤ 2

Q. For any complex numbers $z_{1}$ and $z_{2}$ , the maximum value of $\frac{z _{1} z ˉ _{2} + z ˉ _{1} z _{2}}{∣ z _{1} ∣ ∣ z _{2} ∣}$ is

$1/2$

$1$

$3/2$

$2$

Q. For any complex numbers z1​ and z2​, the maximum value of ∣z1​∣∣z2​∣z1​zˉ2​+zˉ1​z2​​ is

1/2

1

3/2

2

∣z1​∣∣z2​∣z1​zˉ2​+zˉ1​z2​​ =2Re(∣z1​z2​∣z1​zˉ2​​) ≤2⋅∣z1​z2​∣∣z1​zˉ2​∣​=2 Alternatively, z1​=r1​eiα and z2​=r2​eiβ ∴∣z1​∥z2​∣z1​zˉ2​+zˉ1​z2​​ =ei(α−β)+e−i(α−β) =2cos(α−β) ≤2

Q. For any complex numbers $z_{1}$ and $z_{2}$ , the maximum value of $\frac{z _{1} z ˉ _{2} + z ˉ _{1} z _{2}}{∣ z _{1} ∣ ∣ z _{2} ∣}$ is

$1/2$

$1$

$3/2$

$2$