Find the locus of the points arg((z+1)/(z+2)) = pi

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In summary: Then, according to the definition of a bilinear transformation, the transformation is given by:##z=(\frac{az+b}{cz+d})##The above equation is a map from the real plane into the complex plane. By looking at the coordinates of the points (0,0), (1,0), (0,1), and (1,1), we see that the map is a straight line. But, because the real line is not a circle, we see that the line has an angle of ##\frac{1}{2}## with the horizontal axis. Another way to say this is that the line passes through the points (0,0), (1,
  • #1
curious__
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Homework Statement
Find the locus of the points arg((z+1)/(z+2)) = pi
Relevant Equations
z = x + iy, realising the fraction of complex numbers, using the arg = pi condition.
I tried the following proof and got -2 < x < -1 and y = 0 but my prof said that there should be something else I am missing. I have no idea what that is. Thank you.

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  • #2
I agree with your final answer but I don't think I agree with your equation for ##w##. It's not immediately clear to me what you were doing there.
 
  • #3
FactChecker said:
I agree with your final answer but I don't think I agree with your equation for ##w##. It's not immediately clear to me what you were doing there.
I set z = x + iy for real numbers x and y, and then did the calculation.
 
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curious__ said:
I set z = x + iy for real numbers x and y, and then did the calculation.
I don't see what's missing.
 
  • #5
I couldn't find an error or another possibility either. I double-checked with WolframAlpha and it only confirmed the result.
 
  • #6
All right. I haven't really thought of verifying with Wolfram Alpha. Yes that should confirm, I don't know, just my professor said that and I wasn't sure. Thank you anyways!
 
  • #7
curious__ said:
I set z = x + iy for real numbers x and y, and then did the calculation.
Of course. Sorry that I did not see that.
You might be interested in another way that I think is simpler:
1) Solve ##w=(z+1)/(z+2)## for ##z##. That gives ##z=(-2w+1)/(w-1)##.
2) Knowing that transformations of this form map straight lines and circles to straight lines and circles, map a few points along the line of arg(##w##)=##\pi## (that is ##w \lt 0##). Include the endpoints of interest to see what line segment or circle arc you get. ( 0 => -1; -1/2 => (2)/(-3/2)=-4/3; -##\infty## => -2)

Transformations ## z => w## of the type ##w=(az+b)/(cz+d)## in the complex plane are called bilinear transformations (Mobius transformations). They have nice properties that make them very convenient to use.
 
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  • #8
Consider when ##x=\pm \infty##
 

Related to Find the locus of the points arg((z+1)/(z+2)) = pi

1. What is the meaning of "locus" in this context?

The term "locus" refers to a set of points that satisfy a given condition or equation. In this case, we are looking for the set of points that satisfy the equation arg((z+1)/(z+2)) = pi.

2. What does "arg" stand for in the equation?

"arg" stands for the argument of a complex number. It represents the angle between the positive real axis and the vector representing the complex number on the complex plane.

3. How do I find the locus of the points in this equation?

To find the locus of the points, we can use algebraic manipulation and geometric interpretation. We can rewrite the equation as arg(z+1) - arg(z+2) = pi and then use the properties of complex numbers to solve for z. We can also plot the points on a complex plane and visually determine the locus.

4. What does it mean for the locus to equal pi?

When the locus equals pi, it means that the angle between the positive real axis and the vector representing the complex number is equal to pi. In other words, the points that satisfy the equation are located on a line that forms a 180-degree angle with the positive real axis.

5. How can I use the locus of the points to solve problems?

The locus of the points can be used to solve various problems in mathematics and physics. For example, it can be used to find the intersection points of two lines or curves, to determine the range of possible values for a given variable, or to analyze the behavior of a system in different scenarios.

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