Loci in the Argand Diagram (9709 Pure Mathematics 3)

Core Concept

Every complex number $z = x + iy$ is represented by the point $P = (x, y)$ in the Argand diagram. The expression $z - a$, where $a$ is a fixed complex number, represents the displacement vector from the point $A$ (representing $a$) to the point $P$ (representing $z$).

The three loci you need are derived directly from this geometric interpretation:

1. Circle locus: $|z - a| = k$

$|z - a|$ is the modulus of $z - a$, i.e. the distance from $P$ to $A$. Setting this equal to a positive constant $k$ gives all points at a fixed distance $k$ from $A$ — a circle with centre $A$ and radius $k$.

The inequality $|z - a| < k$ gives the interior of this circle (excluding the boundary), and $|z - a| \leq k$ includes the boundary.

2. Perpendicular bisector locus: $|z - a| = |z - b|$

The left side is the distance from $P$ to $A$; the right side is the distance from $P$ to $B$. Equality means $P$ is equidistant from $A$ and $B$, so the locus is the perpendicular bisector of the line segment $AB$.

3. Half-line locus: $\arg(z - a) = \alpha$

$\arg(z - a)$ is the argument of the vector from $A$ to $P$, i.e. the angle that $\overrightarrow{AP}$ makes with the positive real direction. Fixing this angle at $\alpha$ gives a half-line (ray) starting at $A$ (not including $A$ itself), making angle $\alpha$ with the positive $x$-direction.