Solve Complex Equation: Find Z, K in \frac{Z-a}{Z-b}=Ke^{±jθ}

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Discussion Overview

The discussion revolves around solving the complex equation \(\frac{Z-a}{Z-b}=Ke^{±jθ}\), where \(Z\) is an unknown complex number, and \(a\), \(b\), and \(K\) are known values. Participants explore the implications of the equation in the complex plane, including the shapes formed by varying \(θ\) and the existence of multiple solutions.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant seeks assistance in solving the equation and describes the geometric interpretations of the solutions in the complex plane based on the value of \(θ\).
  • Another participant provides a formula for \(Z\) in terms of \(a\), \(b\), \(K\), and \(θ\), suggesting a method to find the solution.
  • A participant questions the existence of only one solution derived from the formula, implying that multiple solutions should exist.
  • Another participant points out the presence of the \(\pm\) in the equation, indicating that this should lead to multiple solutions.

Areas of Agreement / Disagreement

Participants express disagreement regarding the number of solutions to the equation. While one participant claims to have found only one solution, others argue that the presence of the \(\pm\) suggests there should be multiple solutions.

Contextual Notes

The discussion does not resolve the conditions under which multiple solutions may exist, nor does it clarify the assumptions involved in the derivation of the proposed formula for \(Z\).

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Hi, I need a little help :smile:

I need to find solution for this equations:

\frac{Z-a}{Z-b}=Ke^{±jθ}

The Z is unknown and it is the complex number. The a and b is known and they are also complex numbers. K is the real number.

I know that for -90^{°}<θ<90^{°} the graph in the complex plane is circle, for -45^{°}<θ<45^{°} the graph in the complex plane is in shape of "tomato" and for -135^{°}<θ<135^{°} is shape of "lens", but I don't know how to solve it.

Sorry if my post is in wrong area.

Thanks for help.
 
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Leaving it to you the conditions of existence:

Z=\frac{a-b.K.\textrm{e}^{ \pm j \theta }}{1-K.\textrm{e}^{ \pm j \theta }}
 
In that way I got only the one solution, where are the other?
For example, let's put b=0, K=1, theta=45°, with above formula we got only the one solution, but there is more than one solution...
 
How do you only get one solution when there's clearly a \pm in his answer?
 

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