Question
The inequality $$\left| {z - 4} \right|{\text{ < }}\left| {z - 2} \right|$$ represents the region given by
A.
$${\text{Re}}\left( z \right) \geqslant 0$$
B.
$${\text{Re}}\left( z \right) < 0$$
C.
$${\text{Re}}\left( z \right) > 0$$
D.
none of these
Answer :
none of these
Solution :
$$\eqalign{
& \left| {z - 4} \right|{\text{ < }}\left| {z - 2} \right| \cr
& \Rightarrow \,\,\left| {\left( {x - 4} \right) + iy} \right| < \left| {\left( {x - 2} \right) + iy} \right| \cr
& \Rightarrow \,\,{\left( {x - 4} \right)^2} + {y^2} < {\left( {x - 2} \right)^2} + {y^2} \cr
& \Rightarrow \,\, - 8x + 16 < - 4x + 4 \cr
& \Rightarrow \,\,4x - 12 > 0 \cr
& \Rightarrow \,\,x > 3 \cr
& \Rightarrow \,\,{\text{Re}}\left( z \right) > 3 \cr} $$