Let S=zC: |z-1|=1 and (√2-1)(z+ z)-i(z- z)=2√2.

Question

Let $S=\{z\in\mathbb{C}: |z-1|=1 	ext{ and } (\sqrt2-1)(z+\bar z)-i(z-\bar z)=2\sqrt2\}$. Let $z_1,z_2\in S$ [be defined by a further geometric condition not fully preserved in the source]. Then [the required quantity] is:

Accepted Answer

Writing $z=x+iy$, the second condition becomes the line $2(\sqrt2-1)x+2y=2\sqrt2$, i.e. $(\sqrt2-1)x+y=\sqrt2$. Intersecting this line with the circle $|z-1|=1$ (centre $(1,0)$, radius 1) determines specific points of $S$; per the source, $z_1$ is the intersection of the circle with a line $L_1$ through a point $P$, and $z_2$ is the intersection of lines $L_1$ and $L_2$ (the shaded-region construction). The final computed value is 2.

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