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ri0t · Oct 21, 2009, 05:02 AM

Find the locus of the following: arg(z-3) - arg(z+2) = pi/6

So far THis is what I have done:

1. I have plotted the two points (on the complex plane)---> (-2,0) and (3,0)

2. Now the locus would go in a circle, with these two points on the circumference.

3. Then I put the angle that is subtended by the chord of these two ponts at the circumference as pi/6

4. The centre of that circle would be (0.5, 0)

5. Hence the angle subtended by the chord at the centre would be pi/3 (angle at centre is twice than angle at circumference)

That is what I got up to. I know my explanation might be hard to visualise So I tried to draw up what I had in word and is attached, please have a look.

Thank You

ri0t

Nhatkiem · Oct 21, 2009, 01:09 PM

Is your drawing suppose to be to scale? Because if it is, those angles are definitely wrong.

ebaines · Oct 21, 2009, 02:00 PM

Hello ri0t. In step 4 - why do you say the center of the circle is at (0.5, 0)? Looks to me like the center is at (0.5, sqrt(3)).

Also - in your drawing you show a full circle, including negatve y values, but I think it's really symmetric about the x axis - in other words, it looks like two partially overlapping circles.

ri0t · Oct 22, 2009, 02:49 AM

Ok, thank you,

soz my step 4 was a mistake, I knew it wasn't at (0.5,0)
I meant for it to say (0.5, y)

so what will be description of the locus?

also I don't get why it reflected about the x-axis?

I don't think the answer can be blow the x-axis as the question says it = pi/6, hence eliminating any negative angles (in complex analysis, the domain of the arg is -pi < theta < pi.

hence if it was below the x-axis it would read : arg(z-3) - arg (z+2) = - pi/6

PS. My diagram was NOT to scale

ebaines · Oct 22, 2009, 05:50 AM

It's been a long time (>30 years!) since I studied complex numbers in grad school, so I am a bit rusty at this. I see your point about the negative y domain - so the locus is just is the portion of the circle with positive values of y.

Do you see why the radius of the circle is 2 and the center is at $(0.5,sqrt(3))$ ?

Chris-infj · Oct 22, 2009, 09:59 PM

ER, I beg to differ. Since the angle subtended by the (invisible) arc AB at the circumference is pi/6 then the angle at the centre of the circle will be pi/3. This means that the radius of the circle will be 5 and not 2. Then, the centre would have coordinates (0.5,y ) as stated in a previous post but then

y = 5 cos (pi/6). That is centre (0.5, 4.33)

And the locus of the complex number z will be anywhere on the edge of c excluding the points A and B.

ri0t · Oct 22, 2009, 10:24 PM

Thanks a lot

That really helped.

BTW. What program did u use for that?

Chris-infj · Oct 22, 2009, 11:10 PM

I used the free geometry package GeoGebra. See GeoGebra

ebaines · Oct 23, 2009, 05:53 AM

Originally Posted by Chris-infj

ER, i beg to differ. ... the radius of the circle will be 5 and not 2.

Yes - my mistake. I'm aftraid that when I was working this I used cordinates (-1,0) and (1,0) as the real axis intercepts instead of (-2,0) and (3,0) to work through the math (it seemed easier that way) - which means my circle is 2/5 the size of the one as stated in the problem, and I forgot to convert back to the dimensions as stated. So yes - radius of 5.