# Thread: Another complex number problem.

1. ## Another complex number problem.

Hello forumers,

While learning complex numbers, i got stuck in this two problems:

1. $\displaystyle z=2-2i\sqrt3$ and $\displaystyle |z+w|=4$, find $\displaystyle w$ if:

a) $\displaystyle z+w$ is imaginary number
b) $\displaystyle z+w$ is real number

2. $\displaystyle z=1+2i.$ Find $\displaystyle w$ if:

$\displaystyle Re({\frac{w}{\overline{z}}})=2$ and $\displaystyle Im(z\cdot \overline{w})=2$.

Thank you.

3. $\displaystyle z+w=i$. If $\displaystyle |z|=\sqrt2$ and If $\displaystyle |w|=\sqrt5$ find complex numbers z and w.

2. 3.
z= a+ib, w=c+id

$\displaystyle |z|^2=a^2+b^2=2$ (1)
$\displaystyle |w|^2=c^2+d^2=5$ (2)

z+w=(a+c)+i(b+d)=i

so a+c=0 and b+d=1
so c=-a and d=1-b

so by (1) and (2)

$\displaystyle a^2+b^2=2$
and
$\displaystyle a^2+(1-b)^2=5$

solving for b;
b=-1

d=1-b=2
$\displaystyle a^2=2-b^2=1$
so a=1 or -1
c=-a=-1 or 1

so z=1-i, w=-1+2i or z=-1-i, w=1+2i

3. Thank you Krahl. I solved 2 remaining problems.

First problem solution is: $\displaystyle w=\frac{\sqrt3}{2}+\frac{1}{2}i$ and $\displaystyle w'=-\frac{\sqrt3}{2}-\frac{1}{2}i$ just for a).
Second problem solutions is: $\displaystyle w=-2-6i$