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  4. The inequality |z-4|< |z-2| represents the region given by,

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Q. The inequality $|z-4|< |z-2|$ represents the region given by,

Complex Numbers and Quadratic Equations

Solution:

Given $ |z-4|^{2}<|z-2|^{2} $
$ \Rightarrow |(x-4)+i y|^{2}<|(x-2)+i y|^{2} $
$ \Rightarrow (x-4)^{2}+y^{2}<(x-2)^{2}+y^{2} $
$ \Rightarrow -4 x<-12 $
$\Rightarrow 4 x>12 ; x>3 $
$ \Rightarrow \text{Re}(z) >3 $