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#.