Conformal mapping, which geometric objects are these?

Question

Given the mapping $$w = \frac{z + i}{z - 1} $$ find the images on the $w$ plane of

$$i) |z - i| = |z - 1|$$ $$ii) x^2 + y^2 = 1$$

---

$$i) |z + i| = |1 + i|$$ $$ii) ???$$

Polar form, maybe? Which geometric objects do these mappings describe?

Answer 1

For each one you need to rearrange and get $z$ as the subject. $$z=\frac{w+i}{w-1}$$ i) substitute for $z$ and simplify and you end up with $$|w+i-1|=1$$ which is a circle centre $1-i$ with radius $1$.

ii) This is equivalent to $|z|^2=1$, so you end up with $$|w+i|=|w-1|$$ which is a straight line, the perpendicular bisector of the points $-i$ and $1$