Question

Consider a non-zero complex number z satisfying the equation $|z+\frac{2}{\overline{z}}|=3$. Then the sum of all possible values of $|z|$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Answer 2

Let $|z|=r$. Since $z$ is a non-zero complex number, $r>0$. We use the property $\overline{z} = \frac{|z|^2}{z} = \frac{r^2}{z}$. Substituting this into the given equation, we get $|z+\frac{2}{r^2/z}| = 3$, which simplifies to $|z(1+\frac{2}{r^2})|=3$. Using the property $|ab|=|a||b|$, we have $|z||1+\frac{2}{r^2}|=3$. Since $r>0$, $1+\frac{2}{r^2}$ is a positive real number, so $|1+\frac{2}{r^2}| = 1+\frac{2}{r^2}$. The equation becomes $r(1+\frac{2}{r^2})=3$. Distributing $r$, we get $r+\frac{2}{r}=3$. Multiplying by $r$ (since $r \neq 0$), we obtain $r^2+2=3r$, which rearranges to the quadratic equation $r^2-3r+2=0$. Factoring this equation gives $(r-1)(r-2)=0$. Thus, the possible values for $r=|z|$ are $1$ and $2$. Both values are positive and thus valid. The sum of all possible values of $|z|$ is $1+2=3$.