CPX HELP!

[soopertaco]

so my teacher put this question up on the board the other day and i was just for some explanation on how exactly to go about do it.

Sketch Arg((z-2)/(z-i)) = pi/4

[Mao]

The locus itself is a circle.

Now applying the restrictions:

implying the circle must lie above this line

implying that the locus must be outside of this disc

Drawing all three and taking the intersection, we end up with the picture attached. The bold line is the locus, the dotted line is the circle before restriction. The two shaded regions are the two restrictions.


Note that , with only when both real and imaginary parts are positive. If these parts are negative, .

[soopertaco]

Thank you so much! one more thing: when you brought arg to the RHS it became tan? how come?
EDIT: never mind i didn't see that note at the end.


I have another question, regarding this question. He didn't come out with a result like yours, instead he said that the numerator was z₁ and the denominator is z₂ and it was like the arguments were being subtracted. so in the end what he sketched was:
arg(z); arg(z-2)=(alpha) and arg(z-i)=(beta) all on the one axis and connected their branches to the endpoint of arg(z) (which makes a triangle with the re(z) axis)

he would then go on to say that (alpha) - (beta) = (delta), where (delta) was pi=4 he then left it like that. With the introduced angles..
I trust your solution but what in the hell is my teacher going on about and i just wanna know if this is in any way shape or form correct?

[kamil9876]

You can turn it into and then approach it like Mao has. I'm assuming arg=Arg here?

[Mao]

What Kamil said. However I can forsee a subtle problem going between , that but , this may or may not be a problem (it might get rid of itself when simplifying, who knows).

I have another question, regarding this question. He didn't come out with a result like yours, instead he said that the numerator was z₁ and the denominator is z₂ and it was like the arguments were being subtracted. so in the end what he sketched was:
arg(z); arg(z-2)=(alpha) and arg(z-i)=(beta) all on the one axis and connected their branches to the endpoint of arg(z) (which makes a triangle with the re(z) axis)

he would then go on to say that (alpha) - (beta) = (delta), where (delta) was pi=4 he then left it like that. With the introduced angles..
I trust your solution but what in the hell is my teacher going on about and i just wanna know if this is in any way shape or form correct?

What you described is logical. We can do this: (with certain subtle restrictions on Arg and arg, of course, but let's not get too technical right now). The train of thought is definitely valid, but I cannot see to what end that will lead to. Specifically I can't exactly see how that will produce the visualization of the locus (unless he plans to work with the compound angle rule for tan, which will end up being a giant cluster-fuck of everything). So my advice would be to ignore it.

[evaever]

NEW QUESTION

find the locus of arg(z+i) - arg(z+1) = pi/2  

z=0 or z=-1/2+(1/rt2 -1/2)i or z=(1/rt2 -1/2) - 1/2 i

[soopertaco]

thank guys; that does take care of the LHS of the problem but its not pi/4 in the question its pi/2 and when i take tan to the other side it's undefined.

[Mao]

NEW QUESTION

find the locus of arg(z+i) - arg(z+1) = pi/2 

z=0 or z=-1/2+(1/rt2 -1/2)i or z=(1/rt2 -1/2) - 1/2 i

Fairly sure it's an arc mate.

[Mao]

thank guys; that does take care of the LHS of the problem but its not pi/4 in the question its pi/2 and when i take tan to the other side it's undefined.

Aha, but , so that problem essentially reduces down to the denominator = 0.

Give that a go, that should give you the right answer.

[Mao]

Out of curiosity, I did the question. Turned out quite difficult.



However, there must be some restrictions, a full circle is too 'easy'. In the above working we didn't really come across anything that yells out 'IM A RESTRICTION', so we dig a bit deeper.

, this implies

describe the angle to horizontal with respect to the (0,-1), describe the angle to horizontal with respect to the (-1,0).
implies if we pick any point, the angle with respect to (0,-1) must be greater than angle with respect to (-1,0). We notice that only the portion of the circle lying above satisfy this criteria.

The other half of the circle is the locus is actually consisted of two parts: for the bottom arc, and for the arc on the left.

[evaever]

NEW QUESTION

find the locus of arg(z+i) - arg(z+1) = pi/2 

z=0 or z=-1/2+(1/rt2 -1/2)i or z=(1/rt2 -1/2) - 1/2 i

Fairly sure it's an arc mate.

Obviously 3 discrete points cannot be a locus. I was merely giving 3 typical points on the locus as a hint.

[Mao]

NEW QUESTION

find the locus of arg(z+i) - arg(z+1) = pi/2 

z=0 or z=-1/2+(1/rt2 -1/2)i or z=(1/rt2 -1/2) - 1/2 i

Fairly sure it's an arc mate.

Obviously 3 discrete points cannot be a locus. I was merely giving 3 typical points on the locus as a hint.

is a locus with two points.

[evaever]

In geometry, a locus is a collection of points which share a property.

A locus may alternatively be described as the path through which a point moves to fulfill a given condition or conditions.

I was referring to the alternative.

My approach is based on circle geometry
[IMG]http://img46.imageshack.us/img46/1104/forumrh.png[/img]

[BubbleWrapMan]

Wouldn't it make more sense to let the real part be zero, and the imaginary part be greater than zero? That's the definition of a number with argument pi/2. You end up with part of a circle, with the boundary y > -x - 1.

[kamil9876]

What Kamil said. However I can forsee a subtle problem going between , that but , this may or may not be a problem (it might get rid of itself when simplifying, who knows).

Yes I agree I posted in a hurry before and cbf'd with checking what exact definition of Arg is used in spec, but using the definition that you provided ( is the angle makes with positive real axis and is in we have :





This is basically because if but  was in the 2nd quadrant then would be in the 3rd quadrant which is not good because then the (absolute value of) Arg is bigger than pi/2 due to the definition used here. Otherwise it is good. (check this yourself).

Basically I prefer to do this by having each line specifying an "equivalent" condition ie(if and only if, ) rather than just bash some algebra and then go fishing for restriction (how will you know that there are no other restrictions in the sea that you may have missed?) but this is easy to say but a pain in the ass to practice (hence my recent laziness a la previous post).

Wouldn't it make more sense to let the real part be zero, and the imaginary part be greater than zero? That's the definition of a number with argument pi/2. You end up with part of a circle, with the boundary y > -x - 1.

Yes I agree this is an efficient and nice way to proceed from Mao's 3rd (but make sure to also take into account what was mentioned here).