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  4. If |z|=1 and |ω-1|=1 where z, ω ∈ C, then the largest set of values of |2 z-1|2+|2 ω-1|2 equals

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Q. If $|z|=1$ and $|\omega-1|=1$ where $z, \omega \in C$, then the largest set of values of $|2 z-1|^2+|2 \omega-1|^2$ equals

Complex Numbers and Quadratic Equations

Solution:

$(2 z-1)(2 \bar{z}-1)+(2 \omega-1)(2 \bar{\omega}-1) $
${\left[4|z|^2-2(z+\bar{z})+1\right]+\left[4|\omega|^2-2(\omega+\bar{\omega})+1\right]}$
$E=6-4 \operatorname{Re}(z)+4 \omega \bar{\omega}-4 \operatorname{Re} \omega $
$\text { Now }|\omega-1|^2=1 \Rightarrow(\omega-1)(\bar{\omega}-1)=1$
$\omega \bar{\omega}-(\omega+\bar{\omega})=0$
$\omega \bar{\omega}=\omega+\bar{\omega} $
$\omega \bar{\omega}=2 \operatorname{Re} \omega $
$\text { Hence } E=6-4 \operatorname{Rez}+4 \operatorname{Re} \omega $
$\text { [Using } \omega \bar{\omega}=2 \operatorname{Re} \omega \text { ] } $
$=6+4(\operatorname{Re} \omega-\operatorname{Re} z)$
$=6+4\left( x _2- x _1\right)$
$E _{\max }=6+4(2+1)=18 $
$E_{\min }=6+4(0-1)=2 \Rightarrow \text { say }[2,18] $