Given Modulus: From the problem: \[ |z_1 - z_0| = \frac{1 - i}{2} = \frac{1}{\sqrt{2}}, \quad |z_2 - z_0| = \sqrt{2}, \, \text{with centre at} \, \left(\frac{1}{2}, \frac{3}{2}\right). \]
Given Points: \[ z_0 = \left(\frac{1}{2}, \frac{3}{2}\right) \quad \text{and} \quad z_1 = (1, 1). \]
Calculate the angle \( \theta \): For \(z_2\), we use: \[ \tan \theta = -1 \quad \implies \quad \theta = 135^\circ. \]
Coordinates of \(z_2\): Using polar coordinates, we find: \[ z_2 = \left(\frac{1}{2} + \sqrt{2}\cos 135^\circ, \frac{3}{2} + \sqrt{2}\sin 135^\circ\right), \] or
\[ z_2 = \left(\frac{1}{2} - \sqrt{2}\cos 135^\circ, \frac{3}{2} - \sqrt{2}\sin 135^\circ\right). \]
Substitute the values: For \( |z_2|^2 \): \[ z_2 = \left(-\frac{1}{2}, \frac{5}{2}\right) \quad \text{or} \quad z_2 = \left(\frac{3}{2}, \frac{1}{2}\right). \]
Result: \[ |z_2|^2 = \frac{26}{4}, \, \frac{5}{2}. \] Minimum value: \[ |z_2|^2_{\text{min}} = \frac{5}{2}. \]
Final Answer: The minimum value of \( |z_2|^2 \) is \(\frac{5}{2}\).