Question

if Z is equal to X + iota Y and U is equal to 1 minus iota Z upon Z + iota if modulus of U is equal to 1 then show that Z is purely real​

Answer 1

∣w∣=1

∣

z−i

1−iz

∣=1

∣1−iz∣=∣z−i∣

∣1−i(x+iy)∣=∣x+i(y−1)∣, where z=x+iy.

(1+y)

2

+(−x)

2

=

x

2

+(y−1)

2

(1+y)

2

+x

2

=x

2

+(y−1)

2

y=0

z=x+i0=x, which is purely real.