Transforming Lines to Circles in the w-plane

[conorordan]

Homework Statement

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

w=\frac{1}{Z-2}

where Z=x+iy and w=u+iv

Show that under T the straight line with equation 2x+y=5 is transformed to a circle in the w-plane with centre \left ( 1,-\frac{1}{2} \right ) and radius \frac{\sqrt{5}}{2}

The Attempt at a Solution

I've worked out that the line 2x+y=5 can be written in locus form as \left|Z-10\right|=\left|Z+10-10i\right|

 

[Joffan]

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

 

[conorordan]

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

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.