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New Member
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Oct 21, 2009, 05:02 AM
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Complex Number - Locus
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
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Junior Member
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Oct 21, 2009, 01:09 PM
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Is your drawing suppose to be to scale? Because if it is, those angles are definitely wrong.
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Expert
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Oct 21, 2009, 02:00 PM
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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.
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New Member
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Oct 22, 2009, 02:49 AM
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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
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Expert
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Oct 22, 2009, 05:50 AM
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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 )) ?
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Junior Member
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Oct 22, 2009, 09:59 PM
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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.
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New Member
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Oct 22, 2009, 10:24 PM
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Thanks a lot
That really helped.
BTW. What program did u use for that?
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Junior Member
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Oct 22, 2009, 11:10 PM
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I used the free geometry package GeoGebra. See GeoGebra
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Expert
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Oct 23, 2009, 05:53 AM
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 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.
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