# Notes - Complex Analysis MT23, Geometry

> Source: https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/notes/geometry/ · Updated: 2023-09-02 · Tags: uni, notes

- [Course - Complex Analysis MT23](https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/)

### Flashcards
Suppose $a, b \in \mathbb C$ are distinct. What does the locus of
$$
|z - a| = |z - b|
$$
represent?::
A line.

What are the two ways of representing a circle in $\mathbb C$ as a locus?::
$$
|z - a| = r
$$
$$
|z - a| = \lambda |z - b|
$$

Circles can be described in $\mathbb C$ as $|z - a| = \lambda |z - b|$. How does this translate to a fact about standard Euclidean geometry?::
The set of all points $P$ such that the ratio of the distances $PA / PB = \lambda$ is a circle.

### Proofs
Prove that every line in $\mathbb C$ can be written as the locus
$$
|z - a| = |z - b|
$$
for some $a, b \in \mathbb C$.::
Todo.

Prove that every circle in $\mathbb C$ can be written as
$$
|z - a| = \lambda |z - b|
$$
for some $a, b \in \mathbb C$ and some $\lambda \in (0, \infty)$.::
Todo.

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