Exam Questions - Loci in the complex plane

Exam Questions - Loci in the complex plane

Understanding Loci in the Complex Plane

• Loci are a set of points satisfying a particular condition.
• In the context of complex numbers, loci can be represented as lines or curves in the complex plane (or Argand diagram).
• Common loci include circles centred on the origin, lines through the origin, and perpendicular bisectors.

• The typical forms of these loci can be expressed as **

  z-a

  =r** for circles, arg(z-a)=θ for lines and **

  z-a

  =

  z-b

  ** for perpendicular bisectors.

Plotting Complex Loci

• To plot a locus, replace ‘z’ with ‘x+iy’ and simplify the equation. This can lead to a standard form for real or imaginary parts of complex numbers or Euclidean distance.

• For example, the circle locus **

  z-a

  =r** becomes **

  (x+iy)-a

  =r** after substitution. Simplifying this leads to (x-a)^2 + y^2 = r^2 which is recognisable equation of a circle in the x-y plane.

• Similar substitution and simplification for line and perpendicular bisector loci lead to their familiar forms.

Intersecting Loci

• Regions of the complex plane can be defined as intersections of simpler loci.
• These regions are shaded areas on an Argand diagram that satisfy all conditions of the intersecting loci.
• To solve these, graph all loci separately then find the common region that lies in all loci.
• It’s important to always interpret the steps clearly, starting from simpler loci and gradually progressing to intersecting ones.

Solving Problems Involving Loci

• Plenty of practice with plotting and understanding loci is key.
• It’s beneficial to get into the habit of cross-checking answers with graphical solutions to ensure consistency.
• Formulate an appropriate strategy to solve a problem, making sure to start from the original condition(s) of the locus, express in terms of ‘z’, then solve like coordinate geometry problems.
• Be comfortable with using both imaginary and real components of the ‘z’ forms.
• Familiarise with the three common loci forms; lines, circles, and perpendicular bisectors and know how to derive them from the original equations.

Challenging practice problems would enable one to pivot between graphical and algebraic approaches, and help them in resolving any difficulties in understanding complex loci.