Question

show that if |z-5i/z+5i|=1 therfore z is a real number

Answer 1

let z = x+iy,
\[ |x + i y - 5i| = |x + i y + 5i| \\
x^2 + (y-5)^2 = x^2 + (y+5)^2 \implies y = 0\]

Answer 2

is this a compltete answer

Answer 3

you need to show that the imaginary part is zero.

Answer 4

see..\[\left| z-5i/z+5i \right|=\left| z-5i \right|/\left| z+5i \right|=1\]
now..\[\left| z-5i \right|=\left| z+5i \right| \]
which means that z is a complex no. which is equi distant from both 5i and -5i...SO Z IS AREAL NUMBER