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Math Help - Complex Loci

  1. #1
    Member
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    May 2008
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    Complex Loci

    Hey I've got two questions (no answers so ones more of just double checking some working!)

    Firstly can someone have a look over this for me:
    Find the loci on the complex z-plane which satisfies
    <br />
z-c=\rho (\frac{1+it}{1-it})

    where c is complex, \rho is real and t is a parameter satisfying -\infty < t < \infty
    my reasoning is this

     z-c=\rho (\frac{(1+it)^2}{(1-it)(1+it)})

     z-c=\rho (\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}i)

    if we change the parameter now by letting  t=\tan \frac{\theta}{2}

    we then have

     z-c=\rho (cos \theta + i\sin \theta)

    so this would be a circle of radius \rho centered on c

    as  |z-c|=|\rho|
    (does this get around the problem that \rho can be negative?)

    or did I screw up somewhere!

    and for the second question any chance of a little hint because I cant see where to start with this
    Find the locus of points in the complex plane which satisfy the following equation
    z=a+bt+ct^2
    where a,b and c are complex and \frac{b}{c} is real
    for this ive worked out that the final condition implys that if each of the numbers a,b,c were written as x_a+y_ai etc.
    then x_cy_b=x_by_c but I cant see a way to apply it so any chance of a hint

    many thanks

    Simon
    Last edited by thelostchild; September 13th 2008 at 04:50 AM.
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  2. #2
    Super Member
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    Oct 2007
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    London / Cambridge
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    Quote Originally Posted by thelostchild View Post
    Hey I've got two questions (no answers so ones more of just double checking some working!)

    Firstly can someone have a look over this for me:


    my reasoning is this

     z-c=\rho (\frac{(1+it)^2}{(1-it)(1+it)})

     z-c=\rho (\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}i)

    if we change the parameter now by letting  t=\tan \frac{\theta}{2}

    we then have

     z-c=\rho (cos \theta + i\sin \theta)

    so this would be a circle of radius \rho centered on c

    as  |z-c|=|\rho|
    (does this get around the problem that \rho can be negative?)

    or did I screw up somewhere!

    and for the second question any chance of a little hint because I cant see where to start with this


    for this ive worked out that the final condition implys that if each of the numbers a,b,c were written as x_a+y_at etc.
    then x_cy_b=x_by_c but I cant see a way to apply it so any chance of a hint

    many thanks

    Simon
    Your solution to the first one is fine. however there is a neater approach to the 1+ti business. say w = 1 + ti then \arg w = \arctan t then it is easier to simplify \frac{w}{w^*}. Also I don't see how P being negative would be a problem.

    part two should be treated as a vector problem.

    firstly b/c is real therefore it may make sense to write c = \alpha b where \alpha is a real constant. so z = a + bt(1+\alpha t). treating a and b as vectors it should be clear that this is the equation of a line. however pay attention to the range of t(1+\alpha t).

    Bobak
    Last edited by bobak; September 13th 2008 at 05:16 AM.
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