Let C be the set of complex numbers.

Question

Let $\mathbb C$ be the set of complex numbers. Let $S_1=\{z\in\mathbb C:|z-2|\le1\}$ and $S_2=\{z\in\mathbb C:z(1+i)+\bar z(1-i)\ge4\}$. Then the maximum value of $\left|z-\dfrac52\right|^2$ for $z\in S_1\cap S_2$ is equal to

Accepted Answer

$S_1$: disk centred $(2,0)$ radius 1. $S_2$: $x-y\ge2$. Points on the boundary circle: $A=(2+\cos\theta,\sin\theta)$, $\theta\in\left[-\dfrac{3\pi}4,\dfrac\pi4\right]$.

$$(AP)^2=\left(\cos\theta-\frac12\right)^2+\sin^2\theta=\frac54-\cos\theta,$$

maximized at $\theta=-\dfrac{3\pi}4$: $(AP)^2=\dfrac54+\dfrac1{\sqrt2}=\dfrac{5+2\sqrt2}4.$

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