write up proof that the circles centered at z= 1, andhaving radius a different f

Question

write up proof that the circles centered at z= 1, andhaving radius a different from 1, all map to circles under theinversion map 1/z.

then complete the proof that this is also true for all circles notpassing through the origin, by imitating the proof i gave forlines, i.e.recall how i went from one line to all lines anduse the same idea. ** please show all your work and graphs..
I worked on, but I'm stuck in somewhere. Below is what I did so far. *** I need to show this can be made into a circle. The last step was turning 1-2u+u^2+v^2=a^2(u^2+v^2) into aformat resembling a circle: v^2 +(u-(1/1-a^2))^2=(a/1-a)^2. Could you explain this for us. and please complete theproof.

Explanation / Answer

1-2u+u^2+v^2 = (au)^2+(av)^2 1-2u+u^2-(au)^2 = (av)^2 - v^2 (1-a^2)(u^2) - 2u +1 = (a^2 -1) v^2 u^2 - (2/(1-a^2))u +1/(1-a^2) = -v^2 then "complete the square wrt "u" (college algebra) u^2 - (2/(1-a^2))u + 1/(1-a^2)^2 - 1/(1-a^2)^2 + 1/(1-a^2) =-v^2 the first three terms above are a perfect square trinomial thus (u - 1/(1-a^2))^2 -1/(1-a^2)^2 + 1/(1-a^2) = -v^2 (u-1/(1-a^2))^2 + v^2= 1/(1-a^2)^2 - 1/(1-a^2) Then just simplify the right hand side with a common denom (u-1/(1-a^2))^2 + v^2= 1/(1-a^2)^2 - (1-a^2)/(1-a^2)^2 (u-1/(1-a^2))^2 + v^2=a^2/(1-a^2)^2 I leave the rest to you

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