Image of circle under linear fractional transform

Question

Given the LFT for a complex $z$, \begin{align*} \phi:z\mapsto \frac{2z+1}{z+2}. \end{align*} I'm asked about the image under $\phi$ of $C:=\left\{\left\lvert z+\frac25\right\rvert = \frac25\right\}$. I've parametrized this as $\gamma: \frac25 \exp(it) - \frac25$ and computed the image \begin{align*} w(t) = 2-\frac{15}{2\exp(it)+8}, \end{align*} but I'm none the wiser from this expression and I also don't see why this doesn't give me a new circle, although $\phi$ is conformal $ad-bc = 4 - 1 = 3 \neq 0$. Any hints?

Answer 1

HINT...Write $w=\frac{2z+1}{z+2}$ and rearrange so you have $$z=\frac {1-2w}{w-2}$$

Then the locus becomes, after some simplification, $$|1-8w|=2|w-2|$$

Putting $w=u+iv$ will give the Cartesian equation of the circle.

Answer 2

Hint: Mobius transformations map generalized circles to generalized circles.

A generalized circle is either a 'regular' circle, or a straight line plus a point at infinity.

Pick three distinct points on the circle you are given, and find their image under your map. It should be evident if the image is a line or a circle.

For example, you can see that your map sends $0$ to $1/2$ and $−4/5$ to $3/2$ and $−2/5+2/5i$ to $4/17+15/34i$. These points definitely lie on a cricle