Q13/099) If /(z - 5i)/ (z +5i)/ = 1 then prove that z is a real number

let z = x + iy

|z + 5i| = | z - 5i|

or | x + iy + 5i | = | x + iy - 5i|

or | x + iy + 5i |^2 = | x + iy - 5i|^2

or x^2 + (y + 5)^2 = x^2 + (y- 5)^2

or (y + 5)^2 = (y- 5)^2 or y^2 + 10y + 25 = y^2 - 10y + 25 => y =0 or z is real