Hello Guys Hope you are Fine
How to determine the Numbers complexes Z such as:
Z, Z (2), 1+Z has the same Module?
Thanks in Advance Guys
So, we have to find $\displaystyle Z$ such as $\displaystyle |Z|=|Z^2|=|Z+1|$.
The first equality can be written as $\displaystyle |Z|=|Z|^2\Rightarrow |Z|=0$ or $\displaystyle |Z|=1$.
If $\displaystyle |Z|=0\Rightarrow Z=0\Rightarrow |Z+1|=1$, but $\displaystyle |Z+1|$ must be 0.
If $\displaystyle |Z|=1=|Z+1|$ and $\displaystyle Z=x+yi$, we have to solve the system:
$\displaystyle \displaystyle\left\{\begin{array}{ll}\sqrt{x^2+y^2 }=1\\\sqrt{(x+1)^2+y^2}=1\end{array}\right.$ or
$\displaystyle \displaystyle\left\{\begin{array}{ll}x^2+y^2=1\\x^ 2+y^2+2x=0\end{array}\right.$
Substracting the first equation from the second we get $\displaystyle x=-\frac{1}{2}$.
Now, plug $\displaystyle x$ in the first equation and solve for $\displaystyle y$.
We get $\displaystyle y=-\frac{\sqrt{3}}{2}$ or $\displaystyle y=\frac{\sqrt{3}}{2}$.
So, there are two complex numbers satisfying the problem:
$\displaystyle Z=-\frac{1}{2}-\frac{\sqrt{3}}{2}i$ and $\displaystyle Z=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$