Sketch curves and regions in the complex plane defined by conditions on modulus and argument, such as circles, perpendicular bisectors, rays and half-planes

What this dot point is asking

NESA wants you to interpret conditions on a complex variable $z$ as geometric objects in the Argand plane. A modulus condition typically gives a circle or its interior; an argument condition gives a ray; an equality of two distances gives a perpendicular bisector. You must sketch the curve or region accurately, including whether boundaries are included.

Modulus conditions: circles and discs

Since $|z - z_0|$ is the distance from $z$ to the fixed point $z_0$, the equation

is the set of points at distance $r$ from $z_0$, a circle of centre $z_0$ and radius $r$. The corresponding inequalities give regions: $|z - z_0| < r$ is the open interior (boundary dashed and excluded), $|z - z_0| \le r$ is the closed disc, and $|z - z_0| > r$ is the exterior.

To handle a condition like $|z - 1 + 2i| = 3$, rewrite the inside as $z - (1 - 2i)$, so the centre is $1 - 2i$ and the radius is $3$.

Equal distances: perpendicular bisectors

The condition

says $z$ is equidistant from the points $a$ and $b$. The locus of such points is the perpendicular bisector of the segment joining $a$ and $b$. To find its Cartesian equation, write $z = x + iy$, square both moduli, and simplify; the squared terms $x^2$ and $y^2$ cancel, leaving a linear equation, confirming a straight line.

Argument conditions: rays

The condition

fixes the angle of the vector from $z_0$ to $z$. The locus is a ray (half-line) starting at $z_0$ and pointing in the direction $\alpha$. The starting point $z_0$ itself is excluded, because $\arg 0$ is undefined; mark it with an open circle. An inequality such as $0 \le \arg(z - z_0) \le \tfrac{\pi}{2}$ describes a wedge-shaped region between two rays.

Combining conditions

Many problems intersect two conditions. For example, $|z| \le 2$ and $0 \le \arg z \le \tfrac{\pi}{3}$ together describe a sector of the disc of radius $2$: a pie slice. Sketch each condition separately, then shade only the overlap. Be precise about which boundary lines or arcs are solid (included) and which are dashed (excluded).