Put a transformation under the form of a rotation in the complex plane

Question

On the complex plane, I have a transformation "T" such that :

$z' = (m+i)z + m - 1 - i$ ($z'$ is the image and $z$ the preimage, $z$ and $z'$ are both complex number)

and $m$ is a real number.

I'd need to determine "$m$" such that this transformation "T" is a rotation.

I know a rotation can be written under the form : $z'- w = k (z - w)$ with "$w$" the complex number associated with the center and "$k$" a complex number modulus 1. But I can't find how to put "T" under the form of a rotation.

Some hint would be very appreciated, Thanks.

Answer 1

Rotations are described by formula $z'-w=k(z-w)$, i.e. $z'=kz+w(1-k)$.

You are given transformation "T" of the form $z'=(m+i)z+m-1-i$. Hence $$ k=m+i $$ Using condition $|k|=1$ you can determine $m$.

The rest is clear.

Answer 2

Use the form you have to get $z'=kz-kw+w=kz+(1-k)w$. Comapre it to the form you have. You know that $k=m+i$ and that $(1-k)w=m-1-i$. Since $k$ has to have modulus $1$, you can find $k$. From there find $w$.