Geometry with Complex Numbers — Circles

A mobile tower at position 2 + 3i on the complex plane has a broadcast radius of 5 km. Which points can receive its signal? Exactly the points z satisfying |z - 2 - 3i| \leq 5 -- every complex number whose distance from 2 + 3i is at most 5. The boundary of that region, |z - 2 - 3i| = 5, is a circle.

That is the entire idea: a circle on the complex plane is the set of all points at a fixed distance from a fixed centre, written as a single modulus equation. From this one-line definition, you can derive a general equation, test whether a point lies inside or outside, and find where circles intersect lines.

The standard equation

Take a concrete case. The circle with centre z_0 = 2 + 3i and radius r = 2 is |z - 2 - 3i| = 2. The point z = 4 + 3i lies on it because |4 + 3i - 2 - 3i| = |2| = 2. The point z = 2 + 5i lies on it because |2 + 5i - 2 - 3i| = |2i| = 2. The point z = 3 + 3i does not lie on it because |3 + 3i - 2 - 3i| = |1| = 1 \neq 2.

Notice: the equation |z - z_0| < r describes the interior of the circle, and |z - z_0| > r describes the exterior. The circle itself is the boundary between inside and outside.

Special case: circle centred at the origin

When z_0 = 0, the equation becomes |z| = r, or x^2 + y^2 = r^2. This is the set of all complex numbers with modulus r. You encountered this in Modulus of Complex Number — the concentric circles of constant modulus are all centred at the origin.

The unit circle |z| = 1 is especially important. It contains 1, -1, i, -i, and every \cos\theta + i\sin\theta. Complex numbers on the unit circle have modulus 1, so their conjugate equals their reciprocal: \bar{z} = 1/z.

The general form

Expand |z - z_0|^2 = r^2:

Write a = -z_0 and b = |z_0|^2 - r^2. Then the equation becomes:

This is the general form of a circle on the complex plane.

Compare this with the general equation of a line: \bar{a}z + a\bar{z} + b = 0. The circle equation has exactly one extra term: z\bar{z}, which equals |z|^2 = x^2 + y^2. A line is the limiting case of a circle as the radius tends to infinity — the z\bar{z} term vanishes (or, more precisely, dividing through by the radius and taking the limit yields a line).

Working with the general form

Here is the practical recipe. Given an equation z\bar{z} + \bar{a}z + a\bar{z} + b = 0:

1. Read off a from the coefficient of \bar{z} (or \bar{a} from the coefficient of z).
2. Centre = -a.
3. Radius = \sqrt{|a|^2 - b}.

For example, take z\bar{z} + (1 - 2i)z + (1 + 2i)\bar{z} - 4 = 0. Here \bar{a} = 1 - 2i, so a = 1 + 2i — wait, check: the coefficient of \bar{z} is 1 + 2i, so a = 1 + 2i and \bar{a} = 1 - 2i. And b = -4.

Centre = -a = -(1 + 2i) = -1 - 2i.

Radius = \sqrt{|1 + 2i|^2 - (-4)} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3.

So this is a circle with centre -1 - 2i and radius 3.

Circle through three points

Three non-collinear points determine a unique circle. Given three points z_1, z_2, z_3 on the circle, the equation z\bar{z} + \bar{a}z + a\bar{z} + b = 0 must be satisfied by each. This gives three equations (each complex equation splits into a pair of real conditions, but since a has two real parameters and b has one, the three equations are just enough to solve for a and b).

The determinant form is:

This is analogous to the determinant condition for collinearity — but with the extra column z\bar{z} that brings in the quadratic term. When the four points are collinear (and no circle passes through them), the determinant degenerates.

Intersection of a circle and a line

Where a line meets a circle is one of the most common problems in complex-plane geometry. The strategy is direct: substitute the line equation into the circle equation.

Suppose the line is \bar{a}z + a\bar{z} + c = 0 (using c for the line's constant to avoid confusion with the circle's a). From the line equation, solve for \bar{z}:

Substitute this into the circle equation |z - z_0|^2 = r^2. Since |z - z_0|^2 = (z - z_0)(\bar{z} - \bar{z}_0), you get a quadratic in z. Solve it. The number of intersection points depends on the discriminant:

• Two intersection points: the line is a secant (it cuts through the circle).
• One intersection point: the line is a tangent (it touches the circle).
• No intersection points: the line misses the circle entirely.

The condition for tangency is that the perpendicular distance from the centre to the line equals the radius.

An interactive exploration

Drag the red point along the circle below. The circle has centre z_0 = 1 + i and radius 2. The readout shows the modulus |z - z_0|, which stays at 2 for every position on the circle.

Worked examples

Common confusions

• "The equation |z - z_0| = r gives a circle only when z_0 is at the origin." Not at all. The centre can be any complex number z_0. The equation |z| = r (circle centred at the origin) is just the special case z_0 = 0.

• "If b < 0 in the general form, there is no circle." The condition for existence is |a|^2 > b, not b > 0. When b is negative, |a|^2 - b = |a|^2 + |b| is always positive, so the circle always exists. A negative b actually guarantees a valid circle.

• "z\bar{z} in the general form equals z^2." No. z\bar{z} = |z|^2 = x^2 + y^2, which is real and non-negative. The quantity z^2 is generally complex. They are different objects.

• "The general form of a circle and a line look the same." Almost — but the circle has the z\bar{z} term (coefficient 1) while the line does not. The presence or absence of z\bar{z} is what distinguishes a circle from a line. When the coefficient of z\bar{z} is zero, you have a line; when it is non-zero, you have a circle.

• "A tangent line meets the circle at two coincident points." This is a matter of interpretation. Algebraically, the quadratic from the substitution has a repeated root (discriminant zero), so there is one point of tangency. Geometrically, the line just grazes the circle. Both descriptions say the same thing.

• "The perpendicular distance from the centre to a secant equals the radius." No — for a secant, the distance is less than the radius. For a tangent, the distance equals the radius. For a line that misses the circle, the distance is greater than the radius.

Going deeper

Circles and Apollonius

The equation \left|\dfrac{z - z_1}{z - z_2}\right| = k, where k > 0 and k \neq 1, describes a circle called the circle of Apollonius. It is the set of all points whose distances from z_1 and z_2 are in a constant ratio k.

When k = 1, the locus degenerates to a straight line — the perpendicular bisector of the segment z_1 z_2. This is the transition case: the "circle of infinite radius." You will see this locus appear naturally in the Loci in Argand Plane article.

Squaring both sides of |z - z_1|^2 = k^2 |z - z_2|^2 and expanding gives a circle equation in the general form. The centre and radius can be extracted using the recipe above.

The power of a point

For a fixed circle |z - z_0| = r and a point p not on the circle, the quantity |p - z_0|^2 - r^2 is called the power of the point p with respect to the circle. It is positive outside the circle, zero on the circle, and negative inside. In the general form, the power of p is simply p\bar{p} + \bar{a}p + a\bar{p} + b — plug the point into the left side of the circle equation.

The power of a point has a remarkable property: if any line through p intersects the circle at points z_1 and z_2, then (p - z_1)\overline{(p - z_2)} + (p - z_2)\overline{(p - z_1)} = 2(|p - z_0|^2 - r^2). This is the algebraic foundation for the radical axis and many intersection results in geometry.

Lines as degenerate circles

On the extended complex plane (the Riemann sphere, where the point \infty is adjoined), a line is a circle that passes through \infty. This elegant unification means that every result about circles on the Riemann sphere automatically includes lines as a special case. Mobius transformations w = (az+b)/(cz+d) map circles-and-lines to circles-and-lines — this is one of their defining properties.

Where this leads next

Circles and lines are the two fundamental curves of the complex plane. The next steps combine them with geometric constraints.

• Loci in Argand Plane — standard locus conditions like |z - z_1| = |z - z_2|, constant argument, and elliptic loci.
• Geometry with Complex Numbers — Lines — the companion article on lines: collinearity, parallelism, perpendicularity, and reflection.
• Modulus of Complex Number — the definition and properties of |z| that underpin the circle equation.
• Circles — the Cartesian treatment of circles, for comparison with the complex approach.
• Geometry with Complex Numbers — Basics — distance, midpoint, and section formula on the complex plane.