Question

If $\left|\frac{z-5i}{z+5i}\right|=1$, prove that $z$ is real.

Answer 1

Let $z=x+iy$ ,where x and y are real numbers.

$\left|\frac{z-5i}{z+5i}\right|=1\Rightarrow |z-5i|=|z+5i|\Rightarrow |x+i(y-5)|=|x+i(y+5)|$

$\because |z|=\sqrt{x^2+y^2}\therefore \sqrt{x^2+(y-5{)}^2}=\sqrt{x^2+(y+5{)}^2}$

On squaring both sides, $x^2+(y-5{)}^2=x^2+(y+5{)}^2\Rightarrow (y-5{)}^2=(y+5{)}^2$

This is true only if y = 0.

$z=x+i\times 0=x$ ,which is a real number, so z is also a real number.