# Thread: complex numbers

1. ## complex numbers

Find the real numbers x and y if $\displaystyle \frac{1}{x+iy}+\frac{1}{1+2i}=1$

I started like this :

Multiply their respective conjugates for both terms :

i got $\displaystyle \frac{x-iy}{x^2+y^2}+\frac{1-2i}{5}=1$
continue solving from here ,

$\displaystyle \frac{x^2+y^2+5x}{5x^2+5y^2}-[\frac{5y+2x^2+2y^2}{5x^2+5y^2}]i=1$

so i ot 2 equations here :

$\displaystyle 5x=4x^2+4y^2$

$\displaystyle 5y+2x^2+2y^2=0$

I am not sure how to continue from here .. not even sure if my above working is correct . THanks .

2. $\displaystyle \left\{\begin{array}{ll}4x^2+4y^2-5x=0\\2x^2+2y^2+5y=0\end{array}\right.$

Multiply the second ecuation by 2 and substract from the first:

$\displaystyle -5x-10y=0\Rightarrow x=-2y$

Replace x in the second equation:

$\displaystyle 10y^2+5y=0$

Then $\displaystyle y_1=0\Rightarrow x_1=0$ which are not good

and $\displaystyle y_2=-\frac{1}{2}\Rightarrow x_2=1$

3. it can be done by this way