1. Complex numbers problem

Let $\displaystyle z$ be a complex number represented by $\displaystyle x +iy$ What is the condition for the relationship between x and y such that $\displaystyle \frac {z}{1+z^2}$ is a real number? A. xy=1 B. x=y C. $\displaystyle x^2 -y^2 =1$ D. $\displaystyle x^2 + y^2 =1$ E. none of them above

2. Re: Complex numbers problem

Originally Posted by shiny718
Let $\displaystyle z$ be a complex number represented by $\displaystyle x +iy$ What is the condition for the relationship between x and y such that $\displaystyle \frac {z}{1+z^2}$ is a real number? A. xy=1 B. x=y C. $\displaystyle x^2 -y^2 =1$ D. $\displaystyle x^2 + y^2 =1$ E. none of them above
I can tell you that the correct answer is D.
If one makes all the conversions to rectangular form, its easy to see that if $\displaystyle y-y^3-x^2y=0$ the fraction is real.
That is the same as $\displaystyle y(1-y^2-x^2)=0$ thus the answer is $\displaystyle x^2+y^2=1$.
It should be noted that if $\displaystyle y=0$ is also a solution.

I will not do the algebra.