Question

\(z_1\) & \(z_2\) are two complex numbers. If the maximum value of \[\frac{z_1\bar z_2+z_2 \bar z_1}{2|z_1||z_2|}\] is \(\lambda\). then find the value of \(5 \lambda\).

Answer 1

@HolsterEmission

Answer 2

If we call \(z_1 = r_1 e^{i \theta_1}\) and \(z_2 = r_2e^{i \theta _2}\) then this problem reduces to finding the maximum of\[\frac{1}2 \left(e^{i(\theta_1-\theta_2)}+e^{i(\theta_2-\theta_1)}\right)\]\[=\frac{1}2 (2\cos(\theta_1 - \theta_2))\]\[= \cos (\theta_1 - \theta_2)\]The maximum value of the above expression is 1, of course.