Describing the graph of a locus diagram on the complex plane

[Inertia_Squared]

Hey there, I'm a Y12 Student doing Extension 2 maths in NSW and I was looking at one question in particular and wondering if I could get some help in strengthening my understanding of the current topic.

The question specifically asks to "Describe the graph of arg((Z-Z₁)/(Z-Z₂)) = a where a is a constant if 0 < a < pi/2."
I know about the arg laws and such for actually representing the locus, I'm coming unstuck with my graph and how it does not appear to match what the answer is describing (unless I'm pulling another stupid mistake, but I don't think I am).

The answer to the question is "Major arc", and though I'm not (entirely) confused by this answer, it certainly wouldn't be my first pick, so I was wondering if anyone would be able to explain to me how the arc is a major arc and why, as I thought that the result would be two minor arc areas separated by pi/2 on one side and pi on the other side.

If some pictures/graphs are needed to further clarify my meaning feel free to ask, the question and solutions themselves provide no graph/drawing, however, simply the question and answer stated above.

I'm marking this as a question, though I wouldn't mind an open discussion on the topic as well, I would like to learn as much as I possibly can!

Thanks for your time reading this far, and I appreciate all the help I can get!

 

[Dr.Peterson]

Hey there, I'm a Y12 Student doing Extension 2 maths in NSW and I was looking at one question in particular and wondering if I could get some help in strengthening my understanding of the current topic.

The question specifically asks to "Describe the graph of arg((Z-Z₁)/(Z-Z₂)) = a where a is a constant if 0 < a < pi/2."
I know about the arg laws and such for actually representing the locus, I'm coming unstuck with my graph and how it does not appear to match what the answer is describing (unless I'm pulling another stupid mistake, but I don't think I am).

The answer to the question is "Major arc", and though I'm not (entirely) confused by this answer, it certainly wouldn't be my first pick, so I was wondering if anyone would be able to explain to me how the arc is a major arc and why, as I thought that the result would be two minor arc areas separated by pi/2 on one side and pi on the other side.

If some pictures/graphs are needed to further clarify my meaning feel free to ask, the question and solutions themselves provide no graph/drawing, however, simply the question and answer stated above.

I'm marking this as a question, though I wouldn't mind an open discussion on the topic as well, I would like to learn as much as I possibly can!

Thanks for your time reading this far, and I appreciate all the help I can get!

In order to see if you have made a mistake, I'd like to see your work. How did you come to your conclusion? Did you draw a graph, perhaps of an example case, that looks as you describe? And what are the "arg laws" you refer to?

 

[Inertia_Squared]

Here is the graph for the question I drew (sorry for the sideways image) by arg laws I just meant that [arg(a/b) = arg(a) - arg(b)]. I came to my conclusion by assuming that the area specified is a region due to an inequality being present, and it appears to want the area shaded when the angle between is between 0 and pi/2, the centre dot with a strike through the centre indicates two arbitrarily close points of origin for each ray (the arrows). If you'd like there are a few subquestions (you can see a portion of them above and below the question) I could show my interpretations of those and the resulting answers as well in case there is some consistent error I am making.

There isn't any working beyond this as its a graphical question, though if it would assist you I can re-do the question with a step-by-step on my process through the problem, I'd just like to avoid this if necessary as I don't have too much space left in my exercise book and the next chance I'll get to top up is on Monday (I go through them pretty fast). Worse comes to worse though I can raid the printer supplies

 

[Inertia_Squared]

Thanks Subhotosh for the help with the image formatting

 

[Dr.Peterson]

Here is the graph for the question I drew (sorry for the sideways image) by arg laws I just meant that [arg(a/b) = arg(a) - arg(b)]. I came to my conclusion by assuming that the area specified is a region due to an inequality being present, and it appears to want the area shaded when the angle between is between 0 and pi/2, the centre dot with a strike through the centre indicates two arbitrarily close points of origin for each ray (the arrows). If you'd like there are a few subquestions (you can see a portion of them above and below the question) I could show my interpretations of those and the resulting answers as well in case there is some consistent error I am making.

There isn't any working beyond this as its a graphical question, though if it would assist you I can re-do the question with a step-by-step on my process through the problem, I'd just like to avoid this if necessary as I don't have too much space left in my exercise book and the next chance I'll get to top up is on Monday (I go through them pretty fast). Worse comes to worse though I can raid the printer supplies

I don't know what your picture means, even with your explanation. Why would there be two "arbitrarily close" points? Where is [MATH]\alpha[/MATH]? And you are not graphing an inequality! The inequality only tells you that [MATH]\alpha[/MATH] is a fixed acute angle.

I think it's clear that you are misunderstanding the problem, somehow.

There are two given points z1 and z2, and you want the locus of a point z satisfying the equation. Draw those two given points (arbitrary), and show a point z; then show where [MATH]\alpha[/MATH] is, and where point z might be other than the point you show, in order for [MATH]\alpha[/MATH] to remain the same.

 

[Inertia_Squared]

If its not too much to ask, would you be able to show me a graph of how you expect the locus to look? If its too much trouble don't worry, but I'd appreciate it if you can!

 

[Dr.Peterson]

If its not too much to ask, would you be able to show me a graph of how you expect the locus to look? If its too much trouble don't worry, but I'd appreciate it if you can!

I want you to do as much of the thinking as you can, so I'm not including the locus; but here is the picture I described:



If $\alpha$ is a constant, and $z_1$ and $z_2$ are fixed, what is the locus of $z$?

 

[Inertia_Squared]

Is there a particular reason why z1 and z2 are fixed points? I thought that they could be at any point so long as they are some arbitrary but equal distance away from z and have an angle of alpha between them, in other words, a partial region of a circle.

 

[Inertia_Squared]

I figured out what I was doing wrong, I had the positions of the points and Z swapped the wrong way (which is kind of embarassing looking back on how obvious it is in the graph you showed ). Either way, thanks so much for the help on clearing it up, I am pretty sure I've got it right now, I'm just uploading this graph and if you think I've got it I'll close this thread. Cheers!

 

[Dr.Peterson]

Is there a particular reason why z1 and z2 are fixed points? I thought that they could be at any point so long as they are some arbitrary but equal distance away from z and have an angle of alpha between them, in other words, a partial region of a circle.

All I can go by is the statement of the problem that you gave:

"Describe the graph of arg((Z-Z1)/(Z-Z2)) = a where a is a constant if 0 < a < pi/2."​

If you were told something more, you need to quote it for us.

This says nothing about being an equal distance away from z; and the locus is presumably the set of all points where z might be, so you can't decide where z1 and z2 are based on where z is.

I figured out what I was doing wrong, I had the positions of the points and Z swapped the wrong way (which is kind of embarassing looking back on how obvious it is in the graph you showed ). Either way, thanks so much for the help on clearing it up, I am pretty sure I've got it right now, I'm just uploading this graph and if you think I've got it I'll close this thread. Cheers!
View attachment 25157

Are you saying the shaded region is the locus, or just the arc?

And are you saying z is a fixed point? Your solution, whatever it is, doesn't really make sense. You need to rethink it.

 

[Inertia_Squared]

I'm pretty sure that was it, but ill check my textbook when I get home, since it was a subquestion and there might’ve been more information.

 

[Inertia_Squared]

No extra information on the textbook but I realised that I showed the wrong graph in my post - this is what I intended to show, with the blue path being the locus. Just to confirm, upon inspection, as Alpha approaches 0 the locus approaches a circle, and as Alpha approaches pi/2 the locus approaches (but doesn't reach, noting the < sign) a semicircle.



I don't believe that the inequality does much besides give a testing range and allow for some greater variety of already infinite points (a fun paradox), and since 'major arc' encompasses all of its values, that's why that is the answer. The radius and distribution of the points are arbitrary, the conditions can still be satisfied with the magnitude of the points being unequal, but in every case the arc produced is a major arc (given the points lie within the angle constraints).

 

[Dr.Peterson]

No extra information on the textbook but I realised that I showed the wrong graph in my post - this is what I intended to show, with the blue path being the locus. Just to confirm, upon inspection, as Alpha approaches 0 the locus approaches a circle, and as Alpha approaches pi/2 the locus approaches (but doesn't reach, noting the < sign) a semicircle.

View attachment 25169

I don't believe that the inequality does much besides give a testing range and allow for some greater variety of already infinite points (a fun paradox), and since 'major arc' encompasses all of its values, that's why that is the answer. The radius and distribution of the points are arbitrary, the conditions can still be satisfied with the magnitude of the points being unequal, but in every case the arc produced is a major arc (given the points lie within the angle constraints).

View attachment 25170

This makes infinitely more sense than what you showed before! Yes, this is the correct locus, and for the right reasons.

The role of the inequality is to distinguish between the major arc ($\alpha<\pi/2$) and the minor arc($\alpha>\pi/2$); it allows you to give a specific answer rather than just "an arc" (though without it, you could state the measure of the arc, which is perhaps even more important).

 

[Inertia_Squared]

Now that the problem is solved, out of curiosity, since the inequality is used to distinguish between the arc type, how would you define a region/shaded-graph for this question, would you need to introduce a modulus and make Z an inequality too?

 

[Dr.Peterson]

Now that the problem is solved, out of curiosity, since the inequality is used to distinguish between the arc type, how would you define a region/shaded-graph for this question, would you need to introduce a modulus and make Z an inequality too?

You'd have to make an entirely different problem. This one has nothing to do with regions, and different kinds of regions would mean different kinds of problems.

I suppose you could ask for the locus of z while allowing $\alpha$ to vary over some interval; but that would not be what a locus usually is, and it wouldn't result in the type of region you drew.

 

[Inertia_Squared]

Got it, thanks!