How to determine the locus of a complex number

Question

we are given that $\Re\left[\frac{z+5}{z-5}\right]=0.$ What is the locus of these points? is it a circle or y axis? my professor told me that it is a circle.

Answer 1

Given $$\Re\left(\frac{z+5}{z-5}\right)=0\;,$$ Now put $z=x+iy\;,$ We get $$\Re\left(\frac{x+5+iy}{x-5+iy}\right)=0\Rightarrow \Re\left[\left(\frac{x+5+iy}{x-5+iy}\right)\cdot \left(\frac{x-5-iy}{x-5-iy}\right)\right]=0$$

So $$\Re\left[\frac{x^2-25+iy(x-5)-iy(x+y)+y^2}{(x-5)^2+y^2}\right]=0$$

So $$\frac{x^2-25+y^2}{(x-5)^2+y^2}=0\Rightarrow x^2+y^2=25$$

Answer 2

$\displaystyle \frac{z+5}{z-5}=-\overline{\left(\frac{z+5}{z-5}\right)}=-\frac{\bar z+5}{\bar z-5}$, as it is purely imaginary.

Multiplying out, we have $(z+5)(\bar z-5)=-(\bar z+5)(z-5)$, or

$z\bar z+5\bar z-5z-25=-z\bar z+5\bar z-5z+25$

Hence $z\bar z=25$, or $|z|=5$, which is clearly a circle.