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Thread: Complex Loci

  1. #1
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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
    $\displaystyle
    z-c=\rho (\frac{1+it}{1-it})$

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

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

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

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

    we then have

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

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

    as $\displaystyle |z-c|=|\rho|$
    (does this get around the problem that $\displaystyle \rho$ 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
    $\displaystyle z=a+bt+ct^2$
    where a,b and c are complex and $\displaystyle \frac{b}{c}$ is real
    for this ive worked out that the final condition implys that if each of the numbers a,b,c were written as $\displaystyle x_a+y_ai$ etc.
    then $\displaystyle x_cy_b=x_by_c$ but I cant see a way to apply it so any chance of a hint

    many thanks

    Simon
    Last edited by thelostchild; Sep 13th 2008 at 04:50 AM.
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  2. #2
    Super Member
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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

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

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

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

    we then have

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

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

    as $\displaystyle |z-c|=|\rho|$
    (does this get around the problem that $\displaystyle \rho$ 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 $\displaystyle x_a+y_at$ etc.
    then $\displaystyle x_cy_b=x_by_c$ 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 $\displaystyle w = 1 + ti $ then $\displaystyle \arg w = \arctan t$ then it is easier to simplify $\displaystyle \frac{w}{w^*}$. Also I don't see how P being negative would be a problem.

    part two should be treated as a vector problem.

    firstly $\displaystyle b/c$ is real therefore it may make sense to write $\displaystyle c = \alpha b$ where $\displaystyle \alpha$ is a real constant. so $\displaystyle z = a + bt(1+\alpha t)$. 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 $\displaystyle t(1+\alpha t)$.

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