Sketch the region of the complex plane

  • Thread starter SteveDC
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  • #1
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Homework Statement



Sketch the region of the complex plane specified by:

|z - 4 + 3i| ≤ 5


Homework Equations





The Attempt at a Solution


I have tried re-writing the modulus as √[(z)^2 (- 4)^2 + (3i)^2] and from this I have managed to arrive at z ≤ 3√2

But not sure if I needed to do this or how I would take it from here in terms of sketching this
 

Answers and Replies

  • #2
tiny-tim
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Hi SteveDC! :smile:
|z - 4 + 3i| ≤ 5

|z - (4 - 3i)| ≤ 5 ? :wink:
 
  • #3
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Sorry, I might need a bigger hint then this! I still don't really understand
 
  • #4
tiny-tim
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how would you draw |z| ≤ 5 ? :wink:
 
  • #5
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As a line along the real axis stretching to a point that is less than or equal to 5?
 
  • #6
pasmith
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Homework Statement



Sketch the region of the complex plane specified by:

|z - 4 + 3i| ≤ 5


Homework Equations





The Attempt at a Solution


I have tried re-writing the modulus as √[(z)^2 (- 4)^2 + (3i)^2]
and from this I have managed to arrive at z ≤ 3√2

That's not how you compute [itex]|z - 4 + 3i|[/itex]. Recall that if [itex]w = a + ib[/itex] then [itex]|w|^2 = a^2 + b^2[/itex].

Set [itex]z = x + iy[/itex] and see what happens.
 
  • #7
HallsofIvy
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In any set in which an absolute value is defined we can interpret |x- y| as the distance between x and y. In particular, in the complex plane, |z- a| is the distance between z and a. If [itex]|z- b|\le r[/itex], for z a variable, b a specific complex number, and r a real number, then z is any point on or inside the circle with center at b and radius r.

(If z is a complex number, [itex]z\le 3\sqrt{2}[/itex] makes no sense. The complex numbers are not an "ordered field".)
 
  • #8
tiny-tim
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(just got back :tongue2:)
how would you draw |z| ≤ 5 ? :wink:
As a line along the real axis stretching to a point that is less than or equal to 5?

ahh … that's where your misunderstandning is …

|z| ≤ 5 is a circle, the circle of all points whose distance from 0 is ≤ 5

i] do you see why that is? (or do you need an explanation?)

ii] now what does |z - i| ≤ 5 look like?
 
  • #9
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Think I've got this now. ii] a circle round the midpoint at i, with radius less than or equal to 5, and z will lie on that radius.
 
  • #10
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Think I've got this now. ii] a circle round the midpoint at i, with radius less than or equal to 5, and z will lie on that radius.

If by "midpoint" you mean "centered", then you are correct.
 
  • #11
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Yep, thanks everyone
 

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