4. Let \(z_{1}, z_{2}, z_{3}\) be three complex numbers on the circle \(|z| = 1\) with \(\arg(z_{1}) = -\pi/4\), \(\arg(z_{2}) = 0\) and \(\arg(z_{3}) = \pi/4\). If \(\left|\, z_{1}\overline{z_{2}} + z_{2}\overline{z_{3}} + z_{3}\overline{z_{1}} \,\right|^{2} = \alpha + \beta\sqrt{2}\), \(\alpha, \beta \in \mathbb{Z}\), then the value of \(\alpha^{2} + \beta^{2}\) is: (1) 24 (2) 41 (3) 31 (4) 29

Key Concepts Needed

1. Modulus–Argument (Polar) Form
   Every complex number on $|z| = 1$ can be written as
   $z = e^{i\theta} = \cos\theta + i\sin\theta.$ Since the modulus is $1$, the conjugate is simply
   $\overline{z} = e^{-i\theta}.$

2. Product with a Conjugate
   $z_a\,\overline{z_b} = e^{i(\theta_a-\theta_b)}.$ The modulus of this product is also $1$ (it is still on the unit circle), and its argument is the difference of angles.

3. Adding Complex Numbers
   Add real parts together, add imaginary parts together.
   Once you have a single complex number $x + iy$, the squared modulus is
   $|x+iy|^2 = x^2 + y^2.$

Logical Chain to Crack the Problem

1. Write each $z_k$ in polar form:
   $z_1 = e^{i(-\pi/4)},\; z_2 = e^{i(0)},\; z_3 = e^{i(\pi/4)}$.
2. Form the three required products:
   $z_1\overline{z_2},\; z_2\overline{z_3},\; z_3\overline{z_1}$.
3. Use angle subtraction:
   $z_1\overline{z_2} = e^{i(-\pi/4 - 0)} = e^{-i\pi/4}$ $z_2\overline{z_3} = e^{i(0 - \pi/4)} = e^{-i\pi/4}$ $z_3\overline{z_1} = e^{i(\pi/4 - (-\pi/4))} = e^{i\pi/2}.$
4. Add the three complex numbers.
   Convert them to $a+ib$ form to add easily.
5. Compute the squared modulus of the sum.
6. Match the numeric result with $\alpha + \beta\sqrt{2}$, read off $\alpha,\beta$, then evaluate $\alpha^2+\beta^2$.

Every step is short because the unit circle turns products into simple angle arithmetic, and $e^{\pm i\pi/4}$ or $e^{i\pi/2}$ convert to neat fractions of $\sqrt2$ or $1$.

Imagine a Clock

1. Picture the unit circle as the clock face. Every number on the rim has length 1 from the centre.
2. A complex number on that rim is like a hand pointing at some angle.
   • $z_1$ points down-right at –45°.
   • $z_2$ points straight right at 0°.
   • $z_3$ points up-right at +45°.
3. We are asked to mix these arrows in a special way, measure how long the new arrow is, and then square that length.
4. Finally, we compare that number with the form (whole number) + (whole number)×√2 and add their squares.

So it’s really about turning angles into simple additions and then using Pythagoras.

Step 1 – Write in Polar Form

Step 2 – Form the Three Products

Step 3 – Convert to Cartesian Form and Add

The third term: $e^{i\pi/2} = i.$ So the sum is

Step 4 – Square of the Modulus

Let $x = \sqrt2$ and $y = 1 - \sqrt2$.

Step 5 – Match with $\alpha + \beta\sqrt{2}$

$\alpha = 5,\quad \beta = -2.$

Step 6 – Compute $\alpha^2 + \beta^2$

$\alpha^2 + \beta^2 = 5^2 + (-2)^2 = 25 + 4 = 29.$

Final Answer

29 [(Option 4)]