Determine the value of $k$ such that the image of circle $|z - 1| = k$ by complex function $f(z) = \dfrac{z-3}{1-2z}$ is a line.
I substitute $z$ in first equation by $f(z)$ then found that every $k$ makes the equation in form of $az + b$ which is a linear equation. Is this true? But I really doubt. Help me
$f$ is a Möbius transformation, and therefore maps a circle to a circle or to a line. The image is a line if and only if some point of the circle is mapped to $\infty$.
In your case $f(\frac 12) = \infty$, so the image is a line if $z = \frac 12$ is on the circle $|z - 1 | = k$. That gives a unique value for $k$.