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Math Help - 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
     w = z(1-z).

    (a). Show that
     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.
    <br />
\begin{array}{lll}<br />
z=x+yi & w = u+vi & w = z-z^2 \\<br />
& & \\<br />
\to u+vi & = &<br />
x+yi - (x+yi)^2 \\<br />
& = & x+yi-(x^2+2xyi-y^2)\\<br />
& = & x+yi-x^2-2xyi+y^2 \\<br />
& = & (x-x^2+y^2)+(y-2xy)i \\<br />
& & \\<br />
\therefore u=x-x^2+y^2 & , & v= y-2xy \\<br />
& & v=y(1-2x)<br />
\end{array}<br />

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

    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.
    <br />
\begin{array}{lll}<br />
z=x+yi & w = u+vi & w = z-z^2 \\<br />
& & \\<br />
\to u+vi & = &<br />
x+yi - (x+yi)^2 \\<br />
& = & x+yi-(x^2+2xyi-y^2)\\<br />
& = & x+yi-x^2-2xyi+y^2 \\<br />
& = & (x-x^2+y^2)+(y-2xy)i \\<br />
& & \\<br />
\therefore u=x-x^2+y^2 & , & v= y-2xy \\<br />
& & v=y(1-2x)<br />
\end{array}<br />
    Yes, that's correct.

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

    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, u= x and v= x- 2x^2. And since x= u, 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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