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Math Help - Complex numbers, how to describe and sketch a locus?

  1. #1
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    Complex numbers, how to describe and sketch a locus?

    I need to describe and sketch a locus:

    Modulus((z+24)/(z-8i))=3

    where do I begin to describe the centre of this locus without converting it into a cartesian equation?

    and, how many ways can the cartesian equation be found?
    I found (x-3)^2+(y-9)^2=90 by using z=x+yi

    if anyone can help it would be much appreciated. Thanks

    Ray
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  2. #2
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    Quote Originally Posted by raayhan View Post
    I need to describe and sketch a locus:

    Modulus((z+24)/(z-8i))=3

    where do I begin to describe the centre of this locus without converting it into a cartesian equation?

    and, how many ways can the cartesian equation be found?
    I found (x-3)^2+(y-9)^2=90 by using z=x+yi

    if anyone can help it would be much appreciated. Thanks

    Ray
    You have |z + 24| = 3|z - 8i|. The locus is a Circle of Apollonius.

    With a bit of effort you can re-arrange the given relation into the form |z - a| = r and hence recognise it as a circle with radius r and centre at z = a. However, I recommend the Cartesian approach of substituting z = x + iy as being more efficient.

    I haven't checked your answer (except to check that you get a circle, which you did) - I assume if you can do basic algebra you will get the correct answer.
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  3. #3
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    Hello, raayhan!

    Your answer is correct! . . . Good work!


    I need to describe and sketch a locus: .|(z+24)/(z-8i)| .= .3

    Where do I begin to describe the centre of this locus
    without converting it into a cartesian equation?
    and how many ways can the cartesian equation be found?

    I found: .(x - 3)^2 + (y - 9)^2 .= /90 . by using z = x + yi

    mr fantastic's description is correct: a Circle of Apollonius.


    We have: .|z + 24| .= .3|z - 8i|

    Let P be an arbitrary point (x, y).

    The equation says:
    . . the distance from P to A(-24,0) is three times its distance from B(0, 8).
    . . . . . . . . . ._______________ . . . . . ____________
    We have: . √(x + 24)^2 + y^2 . = . 3√x^2 + (y-8)^2

    . . . . . x^2 + 48x + 576 + y^2 . = . 9(x^2 + y^2 - 16y + 64)

    . . . . . x^2 + 48x + 576 + y^2 . = . 9x^2 + 9y^2 - 144y + 576

    - - -8x^2 - 48x + 8y^2 - 144y . = . 0

    - - . . . . .x^2 - 6x + y^2 - 18y . = . 0

    x^2 - 6x + 9 + y^2 - 18y + 81 . = . 0 + 9 + 81


    Therefore: . (x - 3)^2 + (y - 9)^2 .= .90

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  4. #4
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    how do I begin to re-arrange into the form |z - a| = r? where do I begin, what is r and a? thanks
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  5. #5
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    Quote Originally Posted by raayhan View Post
    how do I begin to re-arrange into the form |z - a| = r? where do I begin, what is r and a? thanks
    This way of doing it is more difficult than the straightforward method of substituting z = x + iy. I suggest you stick with that method.
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