Minimum value of \( \left|z_{1}+1\right|+\left|z_{2}+1\right|+\mid z_{1} z_{2}+ \) I if \( \left[z_{1} \mid=1\right. \) and \( \left|z_{2}\right|=1 \) is

Question

Minimum value of \( \left|z_{1}+1\right|+\left|z_{2}+1\right|+\mid |z_{1} z_{2}+1 \) I if \( \left[z_{1} \mid=1\right. \) and \( \left|z_{2}\right|=1 \) is

Answer 1

if |z₁| = 1,

Then Z₁ = ±1

if |z₂| = 1,

then z₂ = ±1

the expression is minimum for Z₁ = -1 and z₂ = -1

so,

|-1 + 1| + |- 1 + 1| + |(-1)(-1) + 1| 

= 0 + 0 + 2 

= 2

Comments

z1=cosx+isinx not just +1 or -1