Set of Points in complex plane

Homework Statement

Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1

Homework Equations

|z| = sqrt(x2 + y2)
z = x + iy

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...

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jbunniii
Homework Helper
Gold Member
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 $w$.

Ray Vickson
Homework Helper
Dearly Missed

Homework Statement

Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1

Homework Equations

|z| = sqrt(x2 + y2)
z = x + iy

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...
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 $|z_1 + z_2| \neq |z_1| + |z_2|$.

PeroK
Homework Helper
Gold Member

Homework Statement

Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1

Not sure if I'm approaching this question correctly...
It says "describe" the set of points. Try thinking geometrically.

Mark44
Mentor
It says "describe" the set of points. Try thinking geometrically.
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.