Complex Number---how to show it ?

[scoutfai]

Given that z = x + yi and
w = ( z + 8i )/(z - 6) , z not equal to 6 .
If w is totally imaginary, show that :
x^2 + y^2 + 2x - 48 = 0

i understand the question, but the problem i facing is i only be able to show :
x^2 + y^2 - 6x + 8y = 0
i think that in order to satisfy what the question ask , i need to find y in term of x, but i cant do it.....i dont sure whether is the question wrong already or my mistake. Any expert there, please help.

[Muzza]

w is totally imaginary <=> Re(w) = 0 <=> ...

[matt grime]

I think there are too many continuations of that ellipsis (well, two), so as an aid, have you been taught how to convert division by a complex number into mulitplication by a complex number (one written as real plus i times imaginary)?

1/z = z*/(|z|^2)

now look at the imaginary part

(for muzza the other posibility i thought of involved the argument which didn't seem useful, though that was only a first impression)

[chronon]

Given that z = x + yi and
w = ( z + 8i )/(z - 6) , z not equal to 6 .
If w is totally imaginary, show that :
x^2 + y^2 + 2x - 48 = 0


If z=0 then w=-4i/3 which is totally imaginary, but -48<>0

I think the numerator should be (z + 8)

[TenaliRaman]

yes it should be (z+8)
simply note that (x+8)(x-6) = x^2+2x-48