Represent the set {z : |z - 1| = 2 and Re z less then or equal to 0} on an Argand Diagram.
I have never come across this type of question. Any hints on how to tackle it?
{z : |z - 1| = 2} ..... The distance of z from 1 is always equal to 2. Geometrically then z lies on a circle of radius 2 and centre at z = 1.
{z : Re z less then or equal to 0} ..... You have the left half of the complex plane (the part to the left of the Im(z) axis).
Put the two together and look for the overlap ......
This is a standard problem about loci in the complex plane. Are you familiar with the following ?
- $\displaystyle |z| = r $ describes a circle at the origin of radius r
- $\displaystyle |z- (a +bi)| = r $ describes a circle with centre $\displaystyle (a ,b)$ and radius r
- $\displaystyle |z - (a +bi) | = |z - (c +di) |$ is the perpendicular bisector of the line joining the points $\displaystyle (a ,b)$ and $\displaystyle (c ,d)$
If you know this problem should be very easy, otherwise using the information I gave you attempt the problem and let me know I you have any issues.
Bobak