Complex Analysis MT23, Geometry
Flashcards
Suppose $a, b \in \mathbb C$ are distinct. What does the locus of
\[ \vert z - a \vert = \vert z - b \vert \]
represent?
A line.
What are the two ways of representing a circle in $\mathbb C$ as a locus?
\[ \vert z - a \vert = r\]
\[ \vert z - a \vert = \lambda \vert z - b \vert \]
Circles can be described in $\mathbb C$ as $ \vert z - a \vert = \lambda \vert z - b \vert $. 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
\[ \vert z - a \vert = \vert z - b \vert \]
for some $a, b \in \mathbb C$.
Todo.
Prove that every circle in $\mathbb C$ can be written as
\[ \vert z - a \vert = \lambda \vert z - b \vert \]
for some $a, b \in \mathbb C$ and some $\lambda \in (0, \infty)$.
Todo.