Ultimate Guide to Describing Lines, Curves, and Regions in the Complex Plane

Introduction

In the final section of our Complex Numbers series, we bridge the gap between algebraic equations and visual geometry. Describing Lines, Curves, and Regions requires you to look at a complex equation involving $z$ (like $|z - 2| = 3$ ) and immediately visualize the geometric shape it creates on the Argand diagram. By translating modulus and argument conditions into distances and angles, you will be able to sketch circles, rays, perpendicular bisectors, and shaded regions with ease.

Executive Summary

This guide covers the standard geometric loci in the complex plane:

Circles: $|z - z_0| = r$ represents a circle centered at $z_0$ with radius $r$ .
Perpendicular Bisectors: $|z - z_1| = |z - z_2|$ represents a line exactly halfway between two points.
Rays: $\arg(z - z_0) = \theta$ represents a half-line starting from $z_0$ at an angle $\theta$ .
Regions: Using inequalities (like $\le$ or $>$ ) to shade the interior or exterior of these curves.

What is this about?

In standard coordinate geometry, you learned that $x^2 + y^2 = 9$ represents a circle. But using $x$ and $y$ can become incredibly messy when dealing with complex numbers.

Instead, we use the geometric definitions of modulus and argument.

The modulus $|z - w|$ is simply the distance between point $z$ and point $w$ .
The argument $\arg(z - w)$ is the angle measured from point $w$ to point $z$ .

By interpreting equations through the lens of "distance" and "angle", you bypass pages of algebra and jump straight to the geometric shape.

Main Content

1. Circles

Equation: $|z - z_0| = r$

Geometric Interpretation: "The distance between $z$ and $z_0$ is always exactly $r$ ." A set of points that are always a fixed distance $r$ from a center point $z_0$ forms a circle.

Center: The complex number $z_0$ .
Radius: The real number $r$ .

Example: $|z - (2 + 3i)| = 4$ is a circle centered at $(2, 3)$ with a radius of $4$ . (Careful: $|z + 2| = 4$ means $|z - (-2)| = 4$ , so the center is $-2$ .)

2. Perpendicular Bisectors (Lines)

Equation: $|z - z_1| = |z - z_2|$

Geometric Interpretation: "The distance from $z$ to $z_1$ is equal to the distance from $z$ to $z_2$ ." The set of all points equidistant from two fixed points forms the perpendicular bisector of the line segment joining them. To sketch this, plot $z_1$ and $z_2$ , draw a dotted line between them, and then draw a solid line exactly halfway between them at a $90^\circ$ angle.

Example: $|z - 2| = |z - 4i|$ . The locus is the perpendicular bisector of the line joining $(2,0)$ and $(0,4)$ .

3. Rays (Half-lines)

Equation: $\arg(z - z_0) = \theta$

Geometric Interpretation: "The angle measured from the point $z_0$ to $z$ is exactly $\theta$ ." This creates a straight ray (half-line) starting at $z_0$ and shooting off infinitely in the direction of $\theta$ .

Crucial Rule: The starting point $z_0$ is NOT included in the locus, because the argument of zero is undefined. You must draw an open circle $\circ$ at $z_0$ .

Example: $\arg(z - i) = \frac{\pi}{4}$ . This is a ray starting at $(0,1)$ (with an open circle) pointing up and to the right at a $45^\circ$ angle.

4. Shading Regions

When you replace an equals sign ( $=$ ) with an inequality ( $<, \le, >, \ge$ ), the curve becomes a boundary, and you must shade a region.

Circles: $|z| \le 3$ means "distance is less than or equal to 3". Shade inside the circle (solid boundary). $|z| > 3$ means shade outside (dotted boundary).
Bisectors: $|z - 1| < |z - i|$ means "closer to 1 than to $i$ ". Draw the dotted bisector, and shade the side containing the point $1$ .
Angles: $0 \le \arg(z) \le \frac{\pi}{3}$ . This is a wedge (or sector) of the plane, starting at the origin, sweeping from $0$ radians to $\frac{\pi}{3}$ radians.

Simple Worked Example

Question: Sketch the region in the complex plane defined by $|z - 2| \le 2$ and $0 \le \arg(z) \le \frac{\pi}{4}$ .

Solution: This question requires you to find the intersection of two separate regions.

Step 1: Sketch the circle inequality. $|z - 2| \le 2$ This is a circle centered at $z_0 = 2$ (the point $(2,0)$ ) with a radius of $2$ . Because the radius is 2, the circle touches the origin $(0,0)$ and goes out to $(4,0)$ . The $\le$ symbol means we shade the interior of this circle.

Step 2: Sketch the argument inequalities. $0 \le \arg(z) \le \frac{\pi}{4}$ $\arg(z) = 0$ is the positive real axis. $\arg(z) = \frac{\pi}{4}$ is a ray from the origin at $45^\circ$ . The region between them is the wedge bounded by these two rays.

Step 3: Find the intersection. Draw the circle from Step 1. Draw the wedge from Step 2. Shade the area that is both inside the circle AND inside the wedge. The result looks like a slice of pie cut out of the top half of the circle.

mini-FAQ page

Q: Do I have to prove these geometrically, or can I use algebra? A: The syllabus strongly encourages the geometric approach because it is much faster. However, if you forget a shape, you can always substitute $z = x + iy$ . For example, $|x + iy - 2| = 3 \implies \sqrt{(x-2)^2 + y^2} = 3 \implies (x-2)^2 + y^2 = 9$ . This is clearly the Cartesian equation of a circle!

Q: What if the ray equation is $\arg(z) = \theta + 2\pi$ ? A: Since the argument is defined up to multiples of $2\pi$ , a ray at $\theta$ and a ray at $\theta + 2\pi$ point in the exact same direction. They are the same geometric locus.

Common mistakes to avoid

Forgetting the open circle on a ray: When graphing $\arg(z - z_0) = \theta$ , you must put a hollow dot at $z_0$ . The HSC markers explicitly look for this.
Messing up the signs in the center: The standard form is $|z - z_0|$ . If the question asks for $|z + 3 - i| = 2$ , you must rewrite it as $|z - (-3 + i)| = 2$ . The center is $(-3, 1)$ , not $(3, -1)$ .
Using a solid line for strictly less than ( $<$ ): If the inequality does not have an "or equal to", the boundary curve must be drawn as a dotted/dashed line.

Practice on Vu's Maths Hub

Graphing regions correctly is crucial because these questions often lead into complex geometry proofs involving intersecting circles and chords.

Practice sketching complex loci and shading regions in the HSC Complex Numbers Booklet.
Use circle geometry theorems to prove properties of these intersecting loci in the HSC Last Resorts Booklet.

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