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Loci involving Complex Numbers - NCETM

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									Loci involving Complex Numbers
                  Modulus
• For real numbers, |x| gives the distance of the
  number x from zero on the number line
• For complex numbers, |z| gives the distance
  of the number z from the origin in an Argand
  diagram
• The locus of points representing the complex
  number z, such that |z| = 2 means all points
  2 units from the origin
                     Modulus
• |z - a| gives the distance of z from a
• |z - a| = r gives a circle and |z – a| = |z – b| gives
  a perpendicular bisector
• For more complicated questions, may be easier
  to use |z| = √(x2 + y2)
• |z + 4| = 3|z|
• ↔ (x + 4)2 + y2 = 9(x2 + y2)
• ↔ 8x2 – 8x – 16 + 8y2 = 0
• ↔ (x – ½ )2 + y2 = 9/4
           Modulus Resources
• Flash: Investigation of Loci
• Excel: Spreadsheet Investigating Loci
                 Argument
• If z and w are complex numbers represented
  by points Z and W in the Argand diagram, z-w
  can be represented by the translation from W
  to Z
• Like position vectors in C4, WZ = z – w
• So arg (z – (a + bi)) = θ gives the set of all
  possible translations (vectors) from a +bi in
  the direction given by θ

								
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