Question

If the conjugate of a complex number $z$ is $\frac{1}{z - i}$, then $z$ can be:

• A. $i\left(\frac{1+\sqrt{5}}{2}\right)$
• B. $i\left(\frac{1-\sqrt{5}}{2}\right)$
• C. $\frac{1+i\sqrt{5}}{2}$
• D. $\frac{1-i\sqrt{5}}{2}$

Hint:

Since $|z|^2 - i\bar{z} = 1$ is real, $i\bar{z}$ must be real. This immediately implies that $\bar{z}$ (and thus $z$) has no real part and is purely imaginary, which eliminates options (C) and (D).

Answer 1

The Correct Option is A

Step 1: Concept
We use the fundamental property of complex conjugates: $z\bar{z} = |z|^2$, which is always a real number.

Step 2: Meaning
We are given $\bar{z} = \frac{1}{z - i}$. By rearranging, we can express the imaginary and real parts to restrict the possible form of $z$.

Step 3: Analysis
Multiplying both sides by $z - i$: \[ \bar{z}(z - i) = 1 \implies z\bar{z} - i\bar{z} = 1 \implies |z|^2 - i\bar{z} = 1 \] \[ \implies i\bar{z} = |z|^2 - 1 \] Since $|z|^2 - 1$ is a purely real number, the term $i\bar{z}$ must also be purely real. Let $z = x + iy$, so $\bar{z} = x - iy$: \[ i\bar{z} = i(x - iy) = ix + y \] For $ix + y$ to be real, the imaginary part must be zero: \[ x = 0 \] Thus, $z$ is purely imaginary ($z = iy$). Substituting $z = iy$ and $\bar{z} = -iy$ back into the original relation: \[ -iy = \frac{1}{iy - i} \implies -iy = \frac{1}{i(y - 1)} = \frac{-i}{y - 1} \implies y = \frac{1}{y - 1} \] \[ \implies y^2 - y - 1 = 0 \implies y = \frac{1 \pm \sqrt{5}}{2} \] Thus, $z = iy = i\left(\frac{1 \pm \sqrt{5}}{2}\right)$.

Step 4: Conclusion
Comparing with the options, $z = i\left(\frac{1+\sqrt{5}}{2}\right)$ is a possible value of $z$.

Final Answer: (A)