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Thread: Complex numbers2

  1. #1 Complex numbers2 
    Forum Masters Degree thyristor's Avatar
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    Hi!
    This is a problem taken from a textbook, but it's not a homework problem. I'm just rehearsing for a test.
    Determine which complex numbers satisfy the equation .
    You are supposed to make a geometrical interpretation, and in the key it says "the complex numbers that lie on the ellipse .
    However, I don't understand how they arrive at this result. Could somebody explain, please?


    373 13231-mbm-13231 373
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  3. #2 Re: Complex numbers2 
    Forum Ph.D. Leszek Luchowski's Avatar
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    is the Euclidean distance, in the complex plane, between and Correction: not -1 but -i. Likewise, is the Euclidean distance between and . Correction: not 1 but i.

    Now the locus of points such that the sum of the distances between and two given points is equal to a given constant (greater than the distance between the two points) is an ellipse.

    You do the math to transform the equation to the form shown in your answer. Hint: use the Pythagorean theorem to determine the distances without using the absolute value symbol.

    Good luck, and keep us posted.


    Leszek. Pronounced [LEH-sheck]. The wondering Slav.
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  4. #3 Re: Complex numbers2 
    Forum Masters Degree thyristor's Avatar
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    Quote Originally Posted by Leszek Luchowski
    is the Euclidean distance, in the complex plane, between and . Likewise, is the Euclidean distance between and .

    Now the locus of points such that the sum of the distances between and two given points is equal to a given constant (greater than the distance between the two points) is an ellipse.

    You do the math to transform the equation to the form shown in your answer. Hint: use the Pythagorean theorem to determine the distances without using the absolute value symbol.

    Good luck, and keep us posted.
    I'm sorry if I don't understand, but why would be the distance between z and -1?
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  5. #4  
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    It looks like these are typos. The distances are to -i and i respectively. The original equation means that the sum of the distances to these two points = 3. The geometrical definition of an ellipse is of this form.
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  6. #5  
    Forum Ph.D. Leszek Luchowski's Avatar
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    Yes, I made a typo, or rather, a mistake due to careless writing. Sorry about that, and thanks for the correction.

    Yes the distances are to and .
    Leszek. Pronounced [LEH-sheck]. The wondering Slav.
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  7. #6  
    Forum Masters Degree thyristor's Avatar
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    Ok, so let .
    Our equation is








    But then, if I'm not mistaken, it is wrong in the key, since x corresponds to a and y to b.
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