# Thread: Loci in the Complex Plane

1. ## 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

2. Here's what I have so far:
a.
$
\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. $
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

3. Originally Posted by alexgeek
Here's what I have so far:
a.
$
\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. $
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, $u= x$ and $v= x- 2x^2$. And since $x= u$, $v= x- 2x^2= u- 2u^2$. That is, of course, a parabola.

4. I feel a fool now ha. Thanks for that!