How to Sketch a Locus on the Argand Diagram for a Complex Equation?

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SUMMARY

The discussion focuses on sketching the locus of the complex equation {(z-3+j)/(z-j)} = √5 on the Argand diagram. Participants emphasize the importance of using "i" instead of "j" for clarity. The process involves squaring both sides of the equation and manipulating it to separate real and imaginary components, ultimately leading to a clearer representation of the locus. The equation is transformed into a more manageable form by substituting z with x + iy and expanding the terms.

PREREQUISITES
  • Understanding of complex numbers and their representation on the Argand diagram.
  • Familiarity with the modulus of complex numbers.
  • Knowledge of algebraic manipulation involving complex conjugates.
  • Experience with separating real and imaginary parts of complex equations.
NEXT STEPS
  • Study the properties of complex numbers on the Argand diagram.
  • Learn about the modulus and argument of complex numbers.
  • Explore techniques for manipulating complex equations, including squaring and expanding terms.
  • Investigate graphical methods for representing loci of complex equations.
USEFUL FOR

Mathematicians, engineering students, and anyone interested in complex analysis or graphical representations of complex equations will benefit from this discussion.

dave1987
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given z is complex, sketch the following locus on the Argand diagram:

{(z-3+j)/(z-j) }= square roots of 5

{ }= modulus .

hope anyone can guide .
 
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First, you use "i" instead of "j" like a normal human being, not an engineer!:smile:

"Square" both sides to get
[tex]\frac{(z-3+ i)(\overline{z}-3-i)}{(z+i)(\overline{z}- i)}= 5[/itex]<br /> Setting z= x+ iy that is<br /> [tex](x-2 + i(y+1))(x- 2)- i(y-1)= 5(x+ i(y+1))(x- i(y-1))[/itex]<br /> <br /> Multiply that out and separate real and imaginary terms.[/tex][/tex]
 

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