Complex Numbers

conorordan

1. The problem statement, all variables and given/known data

"The transformation T from the z-plane to the w-plane is given by

[itex]w=\frac{1}{Z-2}[/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 w-plane 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]\left|Z-10\right|=\left|Z+10-10i\right|[/itex]

Joffan

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

conorordan

okay I substituted z into the transformation but I cannot get an equation of a circle to come out, where do I go from here?

Fredrik

Can you find u and v in terms of x? I would do that, and then compute ##(u-1)^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 1-i/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.