All subjects · Further Mathematics · WFM02 Further Pure Mathematics 2

Loci: circles, bisectors and half-lines

Edexcel IAL Further Mathematics · WFM02 Further Pure Mathematics 2

A locus is the set of points that obey a rule. In the Argand diagram |z - a| is the distance from z to the point a, and arg(z - a) is the angle of the line from a to z. These two facts give the three basic loci.

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Circles

|z - a| = r is a circle with centre a and radius r.

Be careful with signs: |z + 1 - 4i| = |z - (-1 + 4i)|, so the centre is (-1, 4).

Perpendicular bisectors

|z - a| = |z - b| means z is the same distance from a and b, so the locus is the perpendicular bisector of the line from a to b.

To get its Cartesian equation, put z = x + iy and square both sides: the x2 and y2 terms cancel, leaving a straight line.

Half-lines

arg(z - a) = θ is a half-line starting at a (but not including a itself) at angle θ to the positive real direction.

For example arg(z - 1 - i) = π/4 is the part of y = x with x > 1.

Converting and sketching

Set z = x + iy and use |x + iy| = √(x2 + y2). When asked to sketch, label the centre and radius, or the points where a line crosses the axes. The June 2025 examiners' report says sketches must give enough detail to identify what is drawn, and notes several sketches with no radius marked.

Worked example

Describe the locus |z - 3 + 2i| = 4 and find its Cartesian equation.

  1. |z - 3 + 2i| = |z - (3 - 2i)|, the distance from z to 3 - 2i.
  2. So the locus is a circle with centre (3, -2) and radius 4.
  3. Cartesian: (x - 3)2 + (y + 2)2 = 16.

Answer: Circle, centre (3, -2), radius 4: (x - 3)2 + (y + 2)2 = 16

Key idea: |z - a| = r: circle centre a. |z - a| = |z - b|: perpendicular bisector of ab. arg(z - a) = θ: half-line from a.

Check yourself

  1. Describe the locus |z + 1 - 4i| = 3.
    • Circle, centre (-1, 4), radius 3
    • Circle, centre (1, -4), radius 3
    • Circle, centre (-1, 4), radius 9
    • Circle, centre (1, 4), radius 3
    Show answer

    Circle, centre (-1, 4), radius 3. z + 1 - 4i = z - (-1 + 4i), so the centre is -1 + 4i.

  2. Find the Cartesian equation of the locus |z - 2| = |z + 4i|.
    • x - 2y + 3 = 0
    • 2x + y + 3 = 0
    • x + 2y + 3 = 0
    • x + 2y - 3 = 0
    Show answer

    x + 2y + 3 = 0. (x - 2)2 + y2 = x2 + (y + 4)2 gives -4x + 4 = 8y + 16, so 4x + 8y + 12 = 0, that is x + 2y + 3 = 0.

  3. Which describes the locus arg(z - 1 - i) = π/4?
    • The half-line y = x for x > 1
    • The whole line y = x
    • The half-line y = x for x < 1
    • The half-line y = x + 1 for x > 0
    Show answer

    The half-line y = x for x > 1. It starts at 1 + i and goes at angle π/4, that is up and to the right along y = x. The point 1 + i itself is excluded.

  4. z lies on the circle |z - 12 - 5i| = 3. Find the greatest value of |z|.
    Show answer

    16. The centre 12 + 5i is 13 from the origin. The farthest point of the circle is 13 + 3 = 16 away.

  5. For the same circle |z - 12 - 5i| = 3, find the least value of |z|.
    Show answer

    10. The nearest point lies on the line from the origin through the centre: 13 - 3 = 10.

  6. The half-line arg z = π/4 meets the locus |z - 4| = |z - 6i| at the point z = p + pi. Find p.
    Show answer

    5. |z - 4| = |z - 6i| gives -8x + 16 = -12y + 36, so 3y - 2x = 5. With y = x: x = 5.

  7. What is the locus |z - a| = |z - b|, where a and b are fixed and different?
    • A circle passing through the two points
    • The perpendicular bisector of the line joining the two points
    • The straight line joining the two points
    • The single midpoint of the two points
    Show answer

    The perpendicular bisector of the line joining the two points. Every point on it is the same distance from a as from b.

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Last updated 8 November 2026. Spotted a mistake? Tell us.