Lesson Plan: Loci in the Complex Plane Using the Modulus | Nagwa Lesson Plan: Loci in the Complex Plane Using the Modulus | Nagwa

Lesson Plan: Loci in the Complex Plane Using the Modulus Mathematics

This lesson plan includes the objectives, prerequisites, and exclusions of the lesson teaching students how to find the loci of a complex equation in the complex plane from the modulus.

Objectives

Students will be able to

  • describe loci of points constrained by conditions on their moduli geometrically,
  • describe loci of points constrained by conditions on their moduli algebraically,
  • apply the fact that the locus of a point 𝑧 that satisfies |𝑧𝑧|=𝑟 is a circle of radius 𝑟,
  • apply the fact that the locus of a point 𝑧 that satisfies |𝑧𝑧|=|𝑧𝑧| is the perpendicular bisector of the line segment joining 𝑧 and 𝑧,
  • apply the fact that the locus of a point 𝑧 that satisfies |𝑧𝑧|=𝑘|𝑧𝑧|, where 𝑘>0, 𝑘1, is a circle,
  • apply the fact that the locus of a point 𝑧 that satisfies |𝑧𝑧|+|𝑧𝑧|=𝑎, where |𝑧𝑧|>𝑎, is an ellipse with foci 𝑧 and 𝑧 with a major axis of length 𝑏.

Prerequisites

Students should already be familiar with

  • complex numbers, in polar and Cartesian form,
  • the complex plane.

Exclusions

Students will not cover

  • transformations of the complex plane,
  • Möbius transformations.

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