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Thread: Loci in the Complex Plane

  1. #1
    Member alexgeek's Avatar
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    Post Loci in the Complex Plane

    Does anyone have any good links for loci in the complex plane? Preferably a video.
    The book I has skims over it a bit too much.

    Also if someone could help me with this question on my paper, I have no idea what to do, not in any rush though.


    The complex numbers z and and w are represented respectively, by points P(x,y) and Q(u,v) in Argand Diagrams and
    $\displaystyle w = z(1-z)$.

    (a). Show that
    $\displaystyle v=y(1-2x)$
    and find an expression for u in terms of x and y.

    (b). The point P moves along the line y=x. Find the Cartesian equation and the locus of Q.


    Diolch yn fawr
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  2. #2
    Member alexgeek's Avatar
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    Here's what I have so far:
    a.
    $\displaystyle
    \begin{array}{lll}
    z=x+yi & w = u+vi & w = z-z^2 \\
    & & \\
    \to u+vi & = &
    x+yi - (x+yi)^2 \\
    & = & x+yi-(x^2+2xyi-y^2)\\
    & = & x+yi-x^2-2xyi+y^2 \\
    & = & (x-x^2+y^2)+(y-2xy)i \\
    & & \\
    \therefore u=x-x^2+y^2 & , & v= y-2xy \\
    & & v=y(1-2x)
    \end{array}
    $

    b. $\displaystyle
    y = x \:\:
    \therefore
    \begin{array}{lll}
    u = x-x^2+x^2 & & v=x(1-2x)\\
    u = x & & v=x-2x^2
    \end{array}
    $

    No idea what now, pretty sure I need to eliminate x and only get things in terms of u and v but have no idea how.

    Cheers
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  3. #3
    MHF Contributor

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    Quote Originally Posted by alexgeek View Post
    Here's what I have so far:
    a.
    $\displaystyle
    \begin{array}{lll}
    z=x+yi & w = u+vi & w = z-z^2 \\
    & & \\
    \to u+vi & = &
    x+yi - (x+yi)^2 \\
    & = & x+yi-(x^2+2xyi-y^2)\\
    & = & x+yi-x^2-2xyi+y^2 \\
    & = & (x-x^2+y^2)+(y-2xy)i \\
    & & \\
    \therefore u=x-x^2+y^2 & , & v= y-2xy \\
    & & v=y(1-2x)
    \end{array}
    $
    Yes, that's correct.

    b. $\displaystyle
    y = x \:\:
    \therefore
    \begin{array}{lll}
    u = x-x^2+x^2 & & v=x(1-2x)\\
    u = x & & v=x-2x^2
    \end{array}
    $

    No idea what now, pretty sure I need to eliminate x and only get things in terms of u and v but have no idea how.

    Cheers
    You've done almost everything! Yes, $\displaystyle u= x$ and $\displaystyle v= x- 2x^2$. And since $\displaystyle x= u$, $\displaystyle v= x- 2x^2= u- 2u^2$. That is, of course, a parabola.
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  4. #4
    Member alexgeek's Avatar
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    I feel a fool now ha. Thanks for that!
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