Question

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!

Answer 1

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`.