Question

If (1 + i)z = (1 – i)^{\(\bar z\)} then prove that z = -i\(\bar z\).

Answer 1

Let z = x + iy

Then, \(\bar z\) = x - iy

Now, Given: (1 + i)z = (1 – i)^{\(\bar z\)}

Therefore,

(1 + i)(x + iy) = (1 – i)(x – iy)

x + iy + xi + i²y = x – iy – xi + i²y

We know that i² = -1, therefore,

x + iy + ix – y = x – iy – ix – y

2xi + 2yi = 0

x = - y

Now, as x = - y

z = - \(\bar z\)

Hence, Proved.