# 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
$\displaystyle w = z(1-z)$.

(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

2. 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

3. Originally Posted by alexgeek
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.

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