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

Question

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

Tardigrade Admin · Accepted Answer

Since z1 and z2 lie on |z|=1 and |z|=2 then |z1|=1 and |z2|=2 |2 z1+z2| ≤ 2|z1|+|z2| ≤ 4 max |2 z1+z2|=4 |z1-z2| ≥|| z1|-| z2||=|1-2|=1 min |z1-z2|=1 |z2+(1/z1)| ≤|z2|+(1/|z1|)=2+1=3 |z2+(1/z1)| ≤ 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 -

$max ∣ 2 z_{1} + z_{2} ∣ = 4$

$min ∣ z_{1} - z_{2} ∣ = 1$

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