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Complex numbers, loci and arc

  1. Dec 15, 2016 #1

    Kajan thana

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    1. The problem statement, all variables and given/known data

    Sketch the loci, find centre point and the radius of the circle.
    args((z-3i)/((z+4))=π/6


    2. Relevant equations
    args(x/y)=args(x)-args(y)
    Circle theorem - inclined angle theorem


    3. The attempt at a solution

    I sketched the circle with major arc.
    Radius= using Pythagorus I got the radius as 5 unit^2 .
    H=O/sinθ . H=2.5/sin(π/6)

    I am stuck on finding the centre point.

     
  2. jcsd
  3. Dec 15, 2016 #2

    mfb

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    Can you find one or two points on the circle?
     
  4. Dec 15, 2016 #3

    Ray Vickson

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    Your radius is wrong.

    Write ##z = x + iy## and express the ratio ##(z-3i)/(z+4)## as ##A(x,y) + i B(x,y)##. How can you get the equation of the curve in terms of the functions ##A(x,y)## and ##B(x,y)##?
     
  5. Dec 16, 2016 #4

    Kajan thana

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    I don't know how to change it into that form.
     
    Last edited: Dec 16, 2016
  6. Dec 16, 2016 #5

    Kajan thana

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    The coordinates are (0,3) and (-4,0)
     
  7. Dec 16, 2016 #6

    mfb

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    With a complex z and c:$$\frac c z = \frac{cz^*}{zz^*}$$
    Here * is the complex conjugation. Now the denominator is real and you can split the fraction into real and imaginary part.
     
  8. Dec 17, 2016 #7

    Kajan thana

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    I finally got the answer right and the radius is 5 unit^2. Your way gave me the same answer as well.
    Thank you so much.
     
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