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Complex Numbers 
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#1
Apr312, 02:02 PM

P: 13

1. The problem statement, all variables and given/known data
"The transformation T from the zplane to the wplane is given by [itex]w=\frac{1}{Z2}[/itex] where [itex]Z=x+iy[/itex] and [itex]w=u+iv[/itex] Show that under T the straight line with equation [itex]2x+y=5[/itex] is transformed to a circle in the wplane with centre [itex]\left ( 1,\frac{1}{2} \right )[/itex] and radius [itex]\frac{\sqrt{5}}{2}[/itex] 3. The attempt at a solution I've worked out that the line [itex]2x+y=5[/itex] can be written in locus form as [itex]\leftZ10\right=\leftZ+1010i\right[/itex] 


#2
Apr312, 02:33 PM

P: 361

##2x+y=5 \implies y=52x## so we're looking for the transform of ##z = x+i(52x)##.



#3
Apr312, 05:12 PM

P: 13




#4
Apr312, 06:39 PM

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P: 9,382

Complex Numbers
Can you find u and v in terms of x? I would do that, and then compute ##(u1)^2+(v+\frac 1 2)^2##. If you get stuck, then show us your work up to the point where you are stuck.
Edit: OK, I actually tried that, and the result I got is kind of a mess. Makes me wonder if the statement you want to prove is actually true. Can you check if you have stated the problem correctly? Edit 2: I tried a couple of specific points on that line (the ones I tried were 2+i and 1+3i), and found that they are mapped to points at the correct distance from 1i/2. So the statement you're supposed to prove is probably OK. This should mean that it's possible to simplify the mess I got to 5/4. Maybe there's a less messy way to do this. It's been a long time since I did one of these problems, so I don't remember if there are any standard tricks. 


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