Let z=a+ib, where a and b are real. If z/(z-i) is real, show that z is imaginary or 0. Help?

Let z=a+ib, where a and b are real. If z/(z-i) is real, show that z is imaginary or 0.

Thanks!

1 Answer
Dec 31, 2017

Here's one method...

Explanation:

Note that:

z/(z-i) = ((z-i)+i)/(z-i) = 1+i/(z-i) = 1+1/(z/i-1)

If this is real then so is 1/(z/i-1) and therefore z/i-1 and therefore z/i.

So if z/i = c for some real number c, then z = ci, which means that z is either pure imaginary or 0.