Question

Find the image of the circle |z+1 | =1 in the complex plane under the
mapping w=1/z​

Answer 1

Answer:

pls mark as brainlist

Step-by-step explanation:

Rewrite w=1/z as z=1/w. I'll do the part (a).

It follows that z−2=1/w−2, and the condition |z−2|=1 implies that

∣∣∣1w−2∣∣∣=1⟹|1−2w|=|w|

(at this point one may refer to Apollonius' theorem to see that this is equation of circle)

Now you may write w=u+iv to see that above is equivalent to

(1−2u)2+4v2=u2+v2⟹3u2+3v2−4u+1=0⟹u2+v2−43u+13=0

and last equation can be written as (u−2/3)2+v2−19=0 which you should recognize as a circle.

You should be able to do part (b) now.