Description and sketching of subsets of the complex plane defined by conditions on modulus and argument, including circles $|z - z_0| = r$, perpendicular bisectors $|z - a| = |z - b|$, rays $\arg(z - z_0) = \alpha$, and the regions defined by the corresponding inequalities

What this dot point is asking

VCAA wants you to take a condition on a complex number $z$, expressed through its modulus or argument, and sketch the set of points it describes on the Argand plane. The standard loci are circles, perpendicular bisectors and rays; inequalities turn these boundaries into regions. The skill is translating algebra into geometry and back.

Circles and discs

The modulus $|z - z_0|$ is the distance from $z$ to the fixed point $z_0$. Setting that distance equal to a constant gives a circle:

The inequalities describe regions: $|z - z_0| < r$ is the open interior (disc without boundary), $|z - z_0| \le r$ is the closed disc, and $|z - z_0| > r$ is the exterior. To verify algebraically, write $z = x + iy$ and $z_0 = a + ib$; then $|z - z_0|^2 = (x - a)^2 + (y - b)^2 = r^2$, the Cartesian circle.

Perpendicular bisectors

The condition $|z - a| = |z - b|$ says $z$ is equidistant from the fixed points $a$ and $b$. Geometrically this is the perpendicular bisector of the line segment joining $a$ and $b$. The corresponding inequality $|z - a| < |z - b|$ is the half plane of points closer to $a$ than to $b$.

Rays and arguments

The argument $\arg(z - z_0)$ is the direction of the displacement from $z_0$ to $z$. Fixing it,

gives a ray: the half line starting at $z_0$ and pointing in the direction $\alpha$. The point $z_0$ itself is excluded, since the argument of $0$ is undefined; draw an open circle there. A pair of inequalities such as $\frac{\pi}{6} \le \arg(z - z_0) \le \frac{\pi}{3}$ describes the angular sector (wedge) between two rays.

Combining conditions

Many questions intersect two or more loci, for example a disc and a sector, and ask for the overlap region. Sketch each locus, shade the region each inequality allows, and take the common part. Mark boundary curves as solid when the inequality is non-strict ($\le$, $\ge$) and dashed when strict ($<$, $>$).

Examples in context

Example 1. $|z| = 3$ is the circle of radius $3$ centred at the origin.

Example 2. $|z - 2| = |z + 2i|$ is the perpendicular bisector of the segment from $2$ to $-2i$, that is, the points equidistant from $(2, 0)$ and $(0, -2)$.

Try this

Q1. Describe the locus $|z - 4i| = 5$. [2 marks]

• Cue. Circle of radius $5$ centred at $(0, 4)$.

Q2. Sketch $\arg(z) = \frac{3\pi}{4}$. [2 marks]

• Cue. Ray from the origin (excluded) into the second quadrant at $135^\circ$.

Q3. Describe the region $1 \le |z| \le 2$. [2 marks]

• Cue. The closed annulus between circles of radius $1$ and $2$ about the origin.