Question

The inequality $|z-4|<|z-2|$ represents the following region .

• A. $Re\left(z\right)>0$
• B. $Re\left(z\right)<0$
• C. $Re\left(z\right)>2$
• D. None of these

Answer 1

The correct option is A $Re\left(z\right)>0$

$|z-4|<|z-2|$

By taking $z=a+ib$

$\Rightarrow |(a-4)+ib|<|(a-2)+ib|$

$\Rightarrow {(a-4)}^2+b^2<{(a-2)}^2+b^2$

$\Rightarrow {(a-4)}^2<{(a-2)}^2$

$\Rightarrow a^2-8a+16<a^2-4a+4$

$\Rightarrow -8a+16<-4a+4$

$\Rightarrow -8a+4a<4-16$

$\Rightarrow -4a<-12$

$\Rightarrow 4a>12$

$\Rightarrow a>\frac{12}{4}=3$

$\Rightarrow a=Re\left(z\right)=3>0$

$\therefore Re\left(z\right)>0$