If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z $ is real, what is $z$?

Question

Two questions from Grade Twelve class on complex numbers

1. If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z$ is real, what is $z$?
2. How do you solve graphically given the modulus of two points? The question is obtain the complex numbers which satisfies $|z|=15$ and $|z-4|=13$.

So far I was able to solve it algebraically but i don't understand how to solve geographically. BTW my answer was $z=-9+i(207)^{0.5}$.

Thanks for all comments and help!

Answer 1

Hints.

(1) Do you know that $z+\bar z$ is always real? If $z+\frac1z$ is also real, then the number $t=(z+\bar z)-(z+\frac1z)=\bar z-\frac1z$ is real, being the difference of two real numbers. So $\bar z=\frac1z+t$ where $t$ is real. Now multiply that last equation by $z$ and use the fact that $z\bar z=|z|^2$. Hmm, the assumption that $z$ is not real hasn't been used yet; you probably need to use that too. You can figure out what $t$ has to be.

(2) $|z-z_0|$ is the distance between $z$ and $z_0$ in the plane; in particular, $|z|=|z-0|$ is the distance between $z$ and the origin. So you are being asked to find the intersection points of two circles.

Answer 2

Here is a hint for the first question. Write $z=a+bi$ with real numbers $a,b$. By assumtion we have $b\neq 0$. Then compute $1/z$ as follows $$ \frac{1}{z}=\frac{a-bi}{(a-bi)(a+bi)}=\frac{a-bi}{a^2+b^2} $$ Now compute $z+\frac{1}{z}$.