Let z1, z2 be two complex numbers represented by points on the circle |z1|=1 and | z 2|=2 respectively, then :

Question

Let z1, z2 be two complex numbers represented by points on the circle |z1|=1 and | z 2|=2 respectively, then : (A)  max |2 z1+z2|=4 (B)  min | z 1- z 2|=1 (C) | z 2+(1/ z 1)| ≤ 3 (D) none of these

Tardigrade Admin · Accepted Answer

|2 z 1+ z 2| ≤ 2| z 1|+| z 2|=2 × 1+2=4 ∴ Maximum value of |2 z 1+ z 2|=4 Clearly | z 1- z 2| is least when 0, z 1, z 2 are collinear. Then | z 1- z 2|=1 Again |z2+(1/z1)| ≤|z2|+|(1/z1)|=2+(1/|z1|)=2+(1/1)=3 ⇒| z 2+(1/ z 1)| ≤ 3

Q. Let $z_{1}, z_{2}$ be two complex numbers represented by points on the circle $∣ z_{1} ∣ = 1$ and $∣ z_{2} ∣ = 2$ respectively, then :

$max ∣ 2 z_{1} + z_{2} ∣ = 4$

$min ∣ z_{1} - z_{2} ∣ = 1$

$∣ ∣ z_{2} + \frac{1}{z _{1}} ∣ ∣ \leq 3$

none of these

Q. Let z1​,z2​ be two complex numbers represented by points on the circle ∣z1​∣=1 and ∣z2​∣=2 respectively, then :

max∣2z1​+z2​∣=4

min∣z1​−z2​∣=1

∣∣​z2​+z1​1​∣∣​≤3

none of these

Q. Let $z_{1}, z_{2}$ be two complex numbers represented by points on the circle $∣ z_{1} ∣ = 1$ and $∣ z_{2} ∣ = 2$ respectively, then :

$max ∣ 2 z_{1} + z_{2} ∣ = 4$

$min ∣ z_{1} - z_{2} ∣ = 1$

$∣ ∣ z_{2} + \frac{1}{z _{1}} ∣ ∣ \leq 3$