Question

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

• A. $\mathrm{Re}(z)\ge 0$
• B. $\mathrm{Re}(z)<3$
• C. $\mathrm{Re}(z)\le 0$
• D. $\mathrm{Re}z>3$

Answer 1

Answer: (D)

If $z$ satisfies $|z-4|=|z-2|$, then $z$ lies on the perpendicular bisector of the segment joining $z=2$ and $z=4$.
i.e., $|z-4|=|z-2|\Rightarrow \mathrm{Re}(z)=3$.
As $z=0$ does not satisfy $|z-4|<|z-2|$, we get $|z-4|<|z-2|$ represents the region $\mathrm{Re}(z)>3$.