Max/Min of |z| and Arg(z) given |z-1+i| <= 1

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In summary, the given equation defines a circle of radius 1 centered at (1,-1), which includes the circumference. The maximum and minimum values of |z| are \sqrt{2}+1 and \sqrt{2}-1, respectively. The extreme values of Arg(z) are -\pi/2 and 0, as determined by the circle touching the positive x-axis and the negative y-axis.
  • #1
danago
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Given that [tex]|z-1+i| \le 1[/tex], find the maximum and minimum value of |z| and Arg(z).

I realize that the equation given defines the interior of a circle of radius 1 centered at (1,-1), which includes the circumference.

For the first part of the question, i am able to represent the equation graphically. From what i understand, |z| is the distance from the origin to any point lying on or within the circle. If this is the case, i can see the minimum and maximum points, but I am not too sure on how to calculate their locations.

For the next part, finding the extreme values of Arg(z), i just read straight from my graph and said that the minimum is [tex]-\pi/2[/tex] and the maximum is 0. Is that right?

Thanks in advance,
Dan.
 
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  • #2
from description, it appears that this is your task:
given the set of points defined by [tex]\{z:|z-1+i|\leq 1\}[/tex] find the complex numbers z such that the vector going from origin to z has max/min length. likewise for angle (so -p1/2 and 0 seem wrong). but then again you said you read straight from your graph, how does your graph look like, or how you derived it? (to help with pin-pointing potential mistakes)EDIT: sorry my mistake
 
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  • #3
I simply drew a circle centered at (1,-1) with radius 1. Both the x and y-axis are tangental to the circle. That pretty much explains what I've drawn.

My book says the argument of a complex number should be defined within -pi to pi, which is how i got 0 and -pi/2, since the circle touches the positive x-axis and the negative y axis.
 
  • #4
Ok, seems like your picture is fine. For |z|, how far is the center of the circle from the origin? Now how far are the closest and farthest points from the center of the circle?
 
  • #5
Ah ok that's a good way to think about it. The distance from the origin to (1,-1) is [tex]\sqrt{2}[/tex], plus another 1 unit (the circles radius) gives a maximum distance of [tex]\sqrt{2}+1[/tex]. The minimum distance will just then be [tex]\sqrt{2}-1[/tex]. Am i right?
 
  • #6
Absolutely.
 
  • #7
Alright thanks for the help :smile: What about the argument of z? Was i right with that?
 
  • #8
Yes, you were.
 
  • #9
Alright, thanks a lot :smile:
 

1. What is the meaning of "Max/Min of |z| and Arg(z) given |z-1+i| <= 1"?

The phrase "Max/Min of |z| and Arg(z) given |z-1+i| <= 1" refers to finding the maximum and minimum values of the absolute value (magnitude) of z and its argument (angle) within a given condition or constraint, specifically when the distance between z and the complex number 1+i is less than or equal to 1.

2. How do you determine the maximum and minimum values of |z| and Arg(z) in this scenario?

To find the maximum and minimum values of |z| and Arg(z) in this situation, you can graph the given constraint |z-1+i| <= 1 on the complex plane and then analyze the points within or on the boundary of the circle to determine the maximum and minimum values. Alternatively, you can use algebraic methods such as substitution and solving for the variables.

3. What is the significance of the complex number 1+i in this problem?

The complex number 1+i is significant in this problem because it serves as the center of the circle defined by the constraint |z-1+i| <= 1. This means that the possible values of z must fall within or on the boundary of this circle in order to satisfy the given condition.

4. Can the maximum and minimum values of |z| and Arg(z) be equal in this scenario?

Yes, it is possible for the maximum and minimum values of |z| and Arg(z) to be equal in this scenario. This occurs when the constraint |z-1+i| <= 1 defines a circle with a radius of 0, meaning that the only valid value for z is 1+i. In this case, both the maximum and minimum values of |z| and Arg(z) will be equal to 1 and 45 degrees, respectively.

5. How can this problem be applied in real-world situations?

The concept of finding the maximum and minimum values of |z| and Arg(z) given a constraint is commonly used in various fields, including physics, engineering, and economics. For example, in physics, this problem can be used to analyze the possible values of a force vector in a given system. In economics, it can be applied to determine the maximum and minimum values of a certain quantity given constraints such as budget or resources.

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