# Set of Points in complex plane

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1. Jan 15, 2015

### monnapomona

1. The problem statement, all variables and given/known data
Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1

2. Relevant equations
|z| = sqrt(x2 + y2)
z = x + iy

3. The attempt at a solution

Tried to put absolute values on every thing by the Triangle inequality
|z| - |1| + |i| = |1|
sqrt(x2 + y2) - 1 + 1 = 1
sqrt(x2 + y2) = 1

Not sure if I'm approaching this question correctly...

2. Jan 15, 2015

### jbunniii

The triangle inequality $|z + w| \leq |z| + |w|$ won't help you, because it's an inequality. The set of points with $|z - 1 + i| = 1$ is not the same as the set of points with $|z| + |1| + |i| = 1$.

A better approach is to recognize that
$$|z - 1 + i| = 1$$
if and only if
$$|z - 1 + i|^2 = 1$$
The squared equation is easier to work with, because for any complex number $w$ we have $|w|^2 = w\overline w$, where $\overline w$ is the complex conjugate of $w$.

3. Jan 15, 2015

### Ray Vickson

No, you are not: $|z - 1 + i| = 1$ does NOT imply that $|z| -|1| + |i| = |1|$ or anything at all like it. In fact, it is almost always true that $|z_1 + z_2| \neq |z_1| + |z_2|$.

4. Jan 15, 2015

### PeroK

It says "describe" the set of points. Try thinking geometrically.

5. Jan 15, 2015

### Staff: Mentor

And along these lines, an important use of the absolute value operation is to indicate the distance between two points. So |z - 1 + i| = 1 could also be written as |z - (1 - i)| = 1.