Does the Möbius map $T(z) = \frac{z - i} {iz + 3}$ maps circle to circle and line to line?

Question

My approach: Let $w = \frac{z-i}{iz + 3}$. Now I wrote $z$ in terms of $w$ i.e $$z = \frac{3w-i}{wi-1}$$ and considered the general equation of circle(line) which is $$ \alpha\mid z \mid ^2 + 2Re(\beta z) + \gamma = 0$$ where $\alpha \neq 0$ (for line $ \alpha=0$) and substituted $z$ in terms of $w$ hoping to find a similar equation for $w$ but expanding in $w$ makes equation look longish and complicated. So is there a simpler concept we can apply to arrive at the answer or is there a simplification possible in calculation?

Answer 1

Hint...you should be able to show by counter-example that not all lines are transformed into lines (nor circles into circles)

For example, take the line $u=1$, where $w=u+iv$. The image of this line is a circle in the $z$ plane...