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For a non-zero complex number $z$, let $f(z)=\frac{1}{\bar{z}}$.

Let $w=f(z)$. As $z$ varies along the line $(1+2i)z-(1-2i)\bar{z}=i$

what curve does $w$ trace?

I have tried by finding out the value of z in terms of $\bar{z}$ and then put it in the function, also the equation of the line is of the form $|z_1-\bar{z_1}|=1$. But still I cannot figure out the image. Any help will be truly appreciated.

Sayantan
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    Hint: substituting $w=\frac{1}{\bar z}$ gives $(-2+i)\bar w + (-2+i)w - w\bar w =0,$. Complete the LHS to a product form and you get the equation of a circle. – dxiv Jun 11 '21 at 20:08

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We can think of the map $f$ as a composition of $z \mapsto \frac{1}{z}$ and $z \mapsto \bar{z}$.

We know that Möbius transformations takes circles to circles. Therefore, the above line is taken to a circle under $z \mapsto \frac{1}{z}$. One way to find out the explicit equation of this circle is to look at the image of any three points on the line and find the equation of the circle passing through those points. $z \mapsto \bar{z}$ is just reflection about the real axis.

Ajay Kumar Nair
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