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CIE A-Level Maths Study Notes

2.9.8 Complex Loci on Argand Diagram

Complex loci on an Argand diagram present a fascinating intersection of algebra and geometry. This comprehensive guide aims to provide a deep understanding of complex loci, their properties, and their applications in solving complex mathematical problems.

Argand Diagram

Image courtesy of Third Space Learning

Loci Basics

  • Loci in complex numbers: Set of points meeting a specific condition.
  • Shown on Argand diagram as complex numbers forming shapes or paths.
  • Key to understanding complex number geometry.

Loci Equations

  • Example: za=k|z - a| = k (with 'zz' as a complex number, 'aa' as a fixed complex number, 'kk' as a positive real number).
  • Represents a circle on Argand diagram, center 'aa', radius 'kk'.
  • Important for visualizing complex number operations.

Loci Inequalities

  • Inequalities, not just equations, define loci.
  • Example: |z - a| < k shows a region inside a circle, center 'aa', radius 'zz'.
  • Crucial for solving problems with regions and boundaries in complex numbers.

Examples

Example 1. Equation: z(3+4i)=5|z - (3 + 4i)| = 5

  • Step 1: z=x+yiz = x + yi, equation becomes (x+yi)(3+4i)=5|(x + yi) - (3 + 4i)| = 5.
  • Step 2: Simplify to (x3)+(y4)i=5|(x - 3) + (y - 4)i| = 5.
  • Step 3: Use modulus, (x3)2+(y4)2=5\sqrt{(x - 3)^2 + (y - 4)^2} = 5.
  • Step 4: Square both sides, (x3)2+(y4)2=25(x - 3)^2 + (y - 4)^2 = 25.
  • Result: Circle with center (3, 4) and radius 5.

Example 2. Inequality: |z + 2 - 3i| < 4

  • Step 1: z=x+yiz = x + yi, inequality becomes |(x + yi) + 2 - 3i| < 4.
  • Step 2: Simplify to |(x + 2) + (y - 3)i| < 4.
  • Step 3: Use modulus, \sqrt{(x + 2)^2 + (y - 3)^2} < 4.
  • Result: Interior of a circle, center (-2, 3), radius 4.

The graph shows these loci:

Argand Diagram
  • The blue circle represents the first equation's locus, a circle with center at (3, 4) and radius 5.
  • The green area represents the second inequality's locus, the interior of a circle centered at (-2, 3) with radius 4.
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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