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Set of Points in complex plane

  • #1

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...
 

Answers and Replies

  • #2
jbunniii
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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
Ray Vickson
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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|##.
 
  • #4
PeroK
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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.
 
  • #5
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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.
 
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